1 00:00:00,000 --> 00:00:05,916 [SQUEAKING] [RUSTLING] [CLICKING] 2 00:00:10,802 --> 00:00:12,010 SCOTT HUGHES: Good afternoon. 3 00:00:12,010 --> 00:00:17,460 So we spent our last lecture laying out 4 00:00:17,460 --> 00:00:20,240 some of the basic foundations, making a couple of definitions. 5 00:00:20,240 --> 00:00:22,500 I want to quickly recap the most important concepts 6 00:00:22,500 --> 00:00:23,940 and definitions. 7 00:00:23,940 --> 00:00:25,947 And then, let me be blunt, I kind of 8 00:00:25,947 --> 00:00:27,780 want to get through these definitions, which 9 00:00:27,780 --> 00:00:31,800 I think it's important to do them precisely, 10 00:00:31,800 --> 00:00:34,560 but there's nothing significantly challenging 11 00:00:34,560 --> 00:00:35,060 about them. 12 00:00:35,060 --> 00:00:37,170 We just need to make sure they are defined very precisely. 13 00:00:37,170 --> 00:00:39,665 So now that you've kind of seen the style of these things, 14 00:00:39,665 --> 00:00:41,790 I would like to sort of move through the next batch 15 00:00:41,790 --> 00:00:44,010 of these definitions quickly enough 16 00:00:44,010 --> 00:00:46,420 that we can start to move into more interesting material. 17 00:00:46,420 --> 00:00:49,780 So a quick recap-- 18 00:00:49,780 --> 00:00:51,910 and my apologies, my daughter has 19 00:00:51,910 --> 00:00:57,880 some kind of a virus that I am desperately trying to make sure 20 00:00:57,880 --> 00:00:59,980 I do not catch. 21 00:00:59,980 --> 00:01:03,247 And so I will be hydrating during this lecture. 22 00:01:03,247 --> 00:01:05,830 So I just want to recap some of the most important concepts we 23 00:01:05,830 --> 00:01:07,340 went over. 24 00:01:07,340 --> 00:01:09,340 So this whole class is essentially 25 00:01:09,340 --> 00:01:12,160 a study in spacetime. 26 00:01:12,160 --> 00:01:14,440 Later, we're going to connect spacetime to gravity. 27 00:01:14,440 --> 00:01:16,148 And general relativity is going to become 28 00:01:16,148 --> 00:01:18,200 the relativistic theory of gravity. 29 00:01:18,200 --> 00:01:20,380 So we began with a fairly mathematical definition. 30 00:01:20,380 --> 00:01:22,120 Spacetime is a manifold of events 31 00:01:22,120 --> 00:01:23,860 that is endowed with a metric. 32 00:01:23,860 --> 00:01:26,320 Manifold, for our purposes, is essentially 33 00:01:26,320 --> 00:01:28,360 just a set in which we understand 34 00:01:28,360 --> 00:01:30,970 how different members of the set are connected to each other. 35 00:01:30,970 --> 00:01:35,680 Events are really just when and where something happens. 36 00:01:35,680 --> 00:01:37,690 We haven't precisely defined metric yet. 37 00:01:37,690 --> 00:01:39,040 We will soon. 38 00:01:39,040 --> 00:01:41,530 But intuitively, just regard it as some kind 39 00:01:41,530 --> 00:01:44,560 of a mathematical object that gives me a notion of distance 40 00:01:44,560 --> 00:01:46,240 between these events. 41 00:01:46,240 --> 00:01:48,970 I tried and I will continue to try 42 00:01:48,970 --> 00:01:50,740 to be very careful to make a distinction 43 00:01:50,740 --> 00:01:52,810 between geometric objects that live 44 00:01:52,810 --> 00:01:55,047 in the manifold, the events themselves 45 00:01:55,047 --> 00:01:57,130 with things like a displacement vector between two 46 00:01:57,130 --> 00:01:59,560 events, other vectors, which I will 47 00:01:59,560 --> 00:02:01,870 recap the definition of in just a moment. 48 00:02:01,870 --> 00:02:06,280 They have an existence and sort of a reality to them 49 00:02:06,280 --> 00:02:09,789 that is deeper and more fundamental 50 00:02:09,789 --> 00:02:12,730 than the representation of that object. 51 00:02:12,730 --> 00:02:17,740 So when I say that the displacement vector is delta t, 52 00:02:17,740 --> 00:02:22,090 delta x, delta y, delta z, that is according to observer O. 53 00:02:22,090 --> 00:02:23,590 And when I write that down-- 54 00:02:23,590 --> 00:02:25,830 I will be very careful as much as possible. 55 00:02:25,830 --> 00:02:27,288 I will occasionally screw this up-- 56 00:02:27,288 --> 00:02:28,840 but I will try to write this down 57 00:02:28,840 --> 00:02:31,150 without using an equal sign. 58 00:02:31,150 --> 00:02:33,640 Equal sign implies a degree of reality 59 00:02:33,640 --> 00:02:36,110 that I do not want to impart to that representation. 60 00:02:36,110 --> 00:02:38,980 So an equal dot with a little o here, 61 00:02:38,980 --> 00:02:41,200 that's my personal notation for delta 62 00:02:41,200 --> 00:02:44,050 x is represented by these components 63 00:02:44,050 --> 00:02:47,620 according to observer O. And for shorthand, I was sometimes just 64 00:02:47,620 --> 00:02:51,570 write this by the collection of indices delta x alpha. 65 00:02:51,570 --> 00:02:56,710 A different observer O bar, they will 66 00:02:56,710 --> 00:02:59,630 represent this this factor using the exact same geometric 67 00:02:59,630 --> 00:03:00,130 object. 68 00:03:00,130 --> 00:03:02,920 They all agree that it's this displacement between two 69 00:03:02,920 --> 00:03:04,660 physical events in spacetime. 70 00:03:04,660 --> 00:03:06,790 But they assign different coordinates to it. 71 00:03:06,790 --> 00:03:08,800 They give a different representation to it. 72 00:03:08,800 --> 00:03:12,972 And we find that representation using a Lorenz transformation. 73 00:03:12,972 --> 00:03:14,680 And I'm not going to write out explicitly 74 00:03:14,680 --> 00:03:16,930 the Lorenz transformation matrix lambda. 75 00:03:16,930 --> 00:03:18,190 I gave it in the last lecture. 76 00:03:18,190 --> 00:03:21,130 And I'm assuming you're all experts in special relativity, 77 00:03:21,130 --> 00:03:23,410 and I don't need to go over that. 78 00:03:23,410 --> 00:03:26,320 So I will then using this as sort 79 00:03:26,320 --> 00:03:29,130 of the prototype, the general notion of a vector 80 00:03:29,130 --> 00:03:30,630 in spacetime, which we'll often call 81 00:03:30,630 --> 00:03:32,380 a 4-vector for the obvious reason 82 00:03:32,380 --> 00:03:34,900 that it has 4 components, I'm going 83 00:03:34,900 --> 00:03:38,557 to treat that as any quartet of numbers that has transformation 84 00:03:38,557 --> 00:03:40,390 properties just like the displacement factor 85 00:03:40,390 --> 00:03:41,098 I just went over. 86 00:03:41,098 --> 00:03:43,270 So if there's some quantity A that 87 00:03:43,270 --> 00:03:46,720 has time like an x or y and z component, 88 00:03:46,720 --> 00:03:50,445 as long as a different observer, for reasons having 89 00:03:50,445 --> 00:03:51,820 to do with the underlying physics 90 00:03:51,820 --> 00:03:53,740 or whatever the heck this A is, as 91 00:03:53,740 --> 00:03:56,410 long as that different observer relates their components 92 00:03:56,410 --> 00:03:59,320 to observer O's components via the Lorenz transformation, 93 00:03:59,320 --> 00:04:01,600 you got yourself a 4-vector. 94 00:04:01,600 --> 00:04:04,958 Any random set of four numbers, that ain't a 4-vector. 95 00:04:04,958 --> 00:04:07,000 You need to have some physics associated with it. 96 00:04:07,000 --> 00:04:08,667 And the physics has to tell it that it's 97 00:04:08,667 --> 00:04:10,830 a thing that's related by a Lorenz transformation. 98 00:04:19,610 --> 00:04:21,110 So I want to pick up this discussion 99 00:04:21,110 --> 00:04:24,410 by introducing four particularly important 100 00:04:24,410 --> 00:04:27,890 and special vectors, which to be honest, 101 00:04:27,890 --> 00:04:29,780 we're not going to use too much beyond some 102 00:04:29,780 --> 00:04:31,970 of the first couple of weeks or so of the course, 103 00:04:31,970 --> 00:04:33,000 but they're very useful. 104 00:04:33,000 --> 00:04:35,458 And one should often bear in mind, even later in the course 105 00:04:35,458 --> 00:04:37,100 when they've sort of disappeared, 106 00:04:37,100 --> 00:04:39,840 that they're coming along for the ride secretly. 107 00:04:39,840 --> 00:04:41,570 And these are basis vectors. 108 00:04:47,210 --> 00:04:49,967 So if I go into frame O-- 109 00:04:49,967 --> 00:04:52,300 I'm just going to pick some particular reference frame-- 110 00:04:59,770 --> 00:05:06,740 I can immediately write down four special vectors. 111 00:05:06,740 --> 00:05:09,400 So remember, this is a Cartesian type of coordinate system. 112 00:05:09,400 --> 00:05:13,390 So I'm going to introduce e0. 113 00:05:13,390 --> 00:05:15,670 I'm going to represent this by just the number 1 114 00:05:15,670 --> 00:05:21,220 in the time slot and 0 everywhere else. 115 00:05:21,220 --> 00:05:28,130 e1, or ex if you prefer, I'm going 116 00:05:28,130 --> 00:05:30,270 to write that down like so. 117 00:05:30,270 --> 00:05:32,490 And you can kind of see where I'm going with this. 118 00:05:48,307 --> 00:05:50,890 The analogy, if you've all seen unit vectors in other classes, 119 00:05:50,890 --> 00:05:52,660 hopefully this is fairly obvious what I'm doing. 120 00:05:52,660 --> 00:05:54,327 I'm just picking out a set of, you know, 121 00:05:54,327 --> 00:05:57,753 little dimensionless simple quantities that point along 122 00:05:57,753 --> 00:05:59,170 the preferred directions that I've 123 00:05:59,170 --> 00:06:01,390 set up in this inertial reference frame. 124 00:06:01,390 --> 00:06:03,070 A compact way of writing this-- 125 00:06:11,210 --> 00:06:14,730 so notice, I have four of these vectors. 126 00:06:14,730 --> 00:06:18,000 And these vectors each have four components. 127 00:06:18,000 --> 00:06:25,790 And so what I can say is that the beta component of unit 128 00:06:25,790 --> 00:06:31,940 vector e alpha is representative, 129 00:06:31,940 --> 00:06:37,080 according to observer O, by delta alpha beta, the Kronecker 130 00:06:37,080 --> 00:06:37,580 delta. 131 00:06:40,550 --> 00:06:43,402 If you're not familiar with this one, 132 00:06:43,402 --> 00:06:45,360 I'm usually reluctant to send you to Wikipedia. 133 00:06:45,360 --> 00:06:46,985 But in this case, I'm going to send you 134 00:06:46,985 --> 00:06:50,790 to Wikipedia, Wiki via Kronecker delta. 135 00:06:50,790 --> 00:06:53,340 Yeah, so what that does is it just emphasizes that 136 00:06:53,340 --> 00:06:54,990 at least in-- by the way, I should 137 00:06:54,990 --> 00:06:58,020 put little o's underneath all of these things, because I 138 00:06:58,020 --> 00:07:01,848 have chosen this according to observer o's representation. 139 00:07:01,848 --> 00:07:04,140 So this is just saying that a coin to observer O, these 140 00:07:04,140 --> 00:07:06,810 are four very special vectors. 141 00:07:06,810 --> 00:07:08,610 The utility of these things-- 142 00:07:14,070 --> 00:07:16,070 up high for everyone in back to be able to see-- 143 00:07:20,210 --> 00:07:28,180 the utility of doing this is that if I now 144 00:07:28,180 --> 00:07:31,720 want to write the vector A as a geometric object, 145 00:07:31,720 --> 00:07:36,190 I can combine the components that observer O uses 146 00:07:36,190 --> 00:07:38,637 with the basis vectors that observer O uses. 147 00:07:38,637 --> 00:07:39,970 And I can sum them all together. 148 00:07:39,970 --> 00:07:41,290 I can put them together. 149 00:07:41,290 --> 00:07:43,020 And then I've got the-- 150 00:07:43,020 --> 00:07:44,380 it's not a representation. 151 00:07:44,380 --> 00:07:45,470 That's a damn vector. 152 00:07:45,470 --> 00:07:45,970 OK? 153 00:07:45,970 --> 00:07:49,750 I put it all together using sort of an internally consistent set 154 00:07:49,750 --> 00:07:52,360 of numbers and basis vectors. 155 00:07:52,360 --> 00:08:01,900 And so I am free to say this, where I actually 156 00:08:01,900 --> 00:08:03,820 use an actual equal sign. 157 00:08:17,282 --> 00:08:18,990 You might stop and think, well, but those 158 00:08:18,990 --> 00:08:21,990 aren't the components of the observer O bar would use. 159 00:08:21,990 --> 00:08:23,010 And you're right. 160 00:08:23,010 --> 00:08:25,530 Observer O bar would not use those components. 161 00:08:25,530 --> 00:08:27,850 They would also not use those basis factors. 162 00:08:27,850 --> 00:08:29,940 We're going to talk about how those things change. 163 00:08:29,940 --> 00:08:35,080 But observer O bar does agree that if they were handed 164 00:08:35,080 --> 00:08:36,960 O's components and O's basis vectors, 165 00:08:36,960 --> 00:08:39,179 this would give me a complete representation of what 166 00:08:39,179 --> 00:08:40,130 the vector is. 167 00:08:40,130 --> 00:08:42,422 OK? 168 00:08:42,422 --> 00:08:44,130 So again, I'm really harping on this sort 169 00:08:44,130 --> 00:08:46,200 of distinction between the geometric object 170 00:08:46,200 --> 00:08:47,560 and the representation. 171 00:08:47,560 --> 00:08:50,075 This is the geometric object. 172 00:08:50,075 --> 00:08:51,792 And we can take advantage of this. 173 00:08:54,390 --> 00:08:56,490 The fact that that combination of things 174 00:08:56,490 --> 00:08:59,550 is the geometric object is a tool that I'm going to now-- 175 00:08:59,550 --> 00:09:02,340 is a fact that I'm going to exploit in order 176 00:09:02,340 --> 00:09:05,460 to figure out how my basis vectors transform 177 00:09:05,460 --> 00:09:06,780 when I change reference frames. 178 00:09:20,660 --> 00:09:29,310 So let me just repeat what I wrote of there. 179 00:09:29,310 --> 00:09:42,610 But-- oops-- good point for just a slight editorial comment. 180 00:09:42,610 --> 00:09:45,130 When you're talking and writing and there 181 00:09:45,130 --> 00:09:47,560 are millions of little sub scripts and indices, 182 00:09:47,560 --> 00:09:49,810 sometimes the brain and the appendages 183 00:09:49,810 --> 00:09:51,790 get out of sync with one another. 184 00:09:51,790 --> 00:09:53,560 I caught that one. 185 00:09:53,560 --> 00:09:54,550 I don't always. 186 00:09:54,550 --> 00:09:55,060 OK? 187 00:09:55,060 --> 00:09:56,720 So if you see something like that 188 00:09:56,720 --> 00:09:59,110 and you kind of go, um, why did that alpha magically turn 189 00:09:59,110 --> 00:10:00,790 into a beta, it's probably a mistake. 190 00:10:00,790 --> 00:10:01,620 Please call it out. 191 00:10:01,620 --> 00:10:02,890 OK? 192 00:10:02,890 --> 00:10:08,050 All right, so this is how I build this geometric object, 193 00:10:08,050 --> 00:10:09,880 using the components and the basis 194 00:10:09,880 --> 00:10:13,582 vectors as measured by O, as used by observer O. 195 00:10:13,582 --> 00:10:15,040 Let's now write out what they would 196 00:10:15,040 --> 00:10:20,000 be if they were measured by a different observer, an O bar 197 00:10:20,000 --> 00:10:20,500 observer. 198 00:10:23,440 --> 00:10:26,500 I know how to get components, the barred components 199 00:10:26,500 --> 00:10:28,030 from the unbarred components. 200 00:10:28,030 --> 00:10:31,900 I don't yet know how to get the barred basis 201 00:10:31,900 --> 00:10:34,040 vectors from the unbarred basis vectors. 202 00:10:34,040 --> 00:10:36,400 But I know that they exist. 203 00:10:36,400 --> 00:10:39,290 And that once I know what they are, this equation is true. 204 00:10:39,290 --> 00:10:40,000 OK? 205 00:10:40,000 --> 00:10:41,680 These are just two different ways 206 00:10:41,680 --> 00:10:43,330 of writing this geometric object, which 207 00:10:43,330 --> 00:10:46,750 every observer agrees has its existence that 208 00:10:46,750 --> 00:10:49,420 transcends the representation. 209 00:10:49,420 --> 00:10:53,100 So let's rewrite what I've got on the right-hand side 210 00:10:53,100 --> 00:10:55,380 of the rightmost equal sign here. 211 00:10:55,380 --> 00:11:08,350 I'm going to write this as lambda mu bar beta A beta 212 00:11:08,350 --> 00:11:09,230 e mu bar. 213 00:11:13,450 --> 00:11:16,657 And then now I'm going to use a trick, which 214 00:11:16,657 --> 00:11:18,490 is what the-- it's not even so much a trick, 215 00:11:18,490 --> 00:11:22,120 but I'm just going to use a fact that is great when you're 216 00:11:22,120 --> 00:11:25,240 working in index notation. 217 00:11:25,240 --> 00:11:28,090 So often when students first encounter 218 00:11:28,090 --> 00:11:30,400 this kind of notation, your instinct 219 00:11:30,400 --> 00:11:32,170 is to try to write everything out using 220 00:11:32,170 --> 00:11:35,110 matrices and things like row vectors and column vectors. 221 00:11:35,110 --> 00:11:37,420 It's a natural thing to do. 222 00:11:37,420 --> 00:11:39,285 I urge you to get over that. 223 00:11:39,285 --> 00:11:40,660 If I get some bandwidth for that, 224 00:11:40,660 --> 00:11:42,160 I'm going to write up a set of notes 225 00:11:42,160 --> 00:11:45,640 this weekend showing how one can translate at least 2 226 00:11:45,640 --> 00:11:47,230 by 2 objects-- 227 00:11:47,230 --> 00:11:49,290 two index objects and one index objects 228 00:11:49,290 --> 00:11:52,960 in a consistent way between matrices and row 229 00:11:52,960 --> 00:11:55,300 vectors and column vectors. 230 00:11:55,300 --> 00:11:57,760 But we're rapidly going to start getting into objects that 231 00:11:57,760 --> 00:12:00,093 are bigger than that, for which trying to represent them 232 00:12:00,093 --> 00:12:02,470 in matrix-like form gets untenable. 233 00:12:02,470 --> 00:12:05,045 In a little while we're going to have a three index object. 234 00:12:05,045 --> 00:12:07,090 And since we don't have three-dimensional chalkboards, 235 00:12:07,090 --> 00:12:09,132 making this sort of matrix representation of that 236 00:12:09,132 --> 00:12:10,862 is challenging. 237 00:12:10,862 --> 00:12:12,820 Soon after that, we'll have four index objects. 238 00:12:12,820 --> 00:12:14,778 And we'll occasionally need to take derivatives 239 00:12:14,778 --> 00:12:17,260 of that four index object, giving us a five index object. 240 00:12:17,260 --> 00:12:19,260 At that point, the ability to sort of treat them 241 00:12:19,260 --> 00:12:21,700 like matrices is hopeless. 242 00:12:21,700 --> 00:12:23,710 So really, you should just be thinking 243 00:12:23,710 --> 00:12:26,368 of this as ordinary multiplication 244 00:12:26,368 --> 00:12:28,660 of the numbers that are represented by these components 245 00:12:28,660 --> 00:12:29,760 as written out here. 246 00:12:29,760 --> 00:12:35,650 And an ordinary multiplication like this, 247 00:12:35,650 --> 00:12:40,960 I can just go ahead and swap the order of multiplication 248 00:12:40,960 --> 00:12:41,780 very easily here. 249 00:12:41,780 --> 00:12:47,550 So what I'm going to do is move the-- 250 00:12:47,550 --> 00:12:48,400 hang on a second. 251 00:12:51,568 --> 00:12:53,360 Yeah, yeah, yeah, now I see what I'm doing. 252 00:12:53,360 --> 00:12:54,068 Sorry about that. 253 00:12:54,068 --> 00:12:58,200 I misread my notes. 254 00:12:58,200 --> 00:13:01,090 I'm just going to move the A onto the other side 255 00:13:01,090 --> 00:13:01,780 of my lambda. 256 00:13:18,560 --> 00:13:20,540 And then I'm going to use the fact 257 00:13:20,540 --> 00:13:24,380 that beta on my right-hand side is a dummy index. 258 00:13:37,880 --> 00:13:40,060 So in that final expression I wrote down over there, 259 00:13:40,060 --> 00:13:41,350 beta is a dummy index. 260 00:13:41,350 --> 00:13:43,600 And I'm free to adjust it to put it into a form that's 261 00:13:43,600 --> 00:13:44,558 more convenient for me. 262 00:14:02,785 --> 00:14:03,410 So let's begin. 263 00:14:03,410 --> 00:14:05,890 Let me just write where I've got this equation right now 264 00:14:05,890 --> 00:14:06,875 over there. 265 00:14:06,875 --> 00:14:12,220 So A alpha E alpha equals-- 266 00:14:12,220 --> 00:14:18,740 and what I've got then is component A beta lambda mu bar 267 00:14:18,740 --> 00:14:19,240 beta-- 268 00:14:23,160 --> 00:14:24,200 keep mu bar. 269 00:14:24,200 --> 00:14:25,070 Sorry about that. 270 00:14:25,070 --> 00:14:27,490 So I'm going to remap my dummy index. 271 00:14:40,260 --> 00:14:43,690 The reason I did that is I can now move this to the other side 272 00:14:43,690 --> 00:14:45,195 and factor out the A alpha. 273 00:15:12,750 --> 00:15:14,730 OK, everyone can see that I hope. 274 00:15:14,730 --> 00:15:16,530 So see what I did was I arranged this 275 00:15:16,530 --> 00:15:18,780 so that I've isolated essentially 276 00:15:18,780 --> 00:15:20,950 only the transformation of the basis vector. 277 00:15:35,300 --> 00:15:37,910 So this equation has to hold no matter 278 00:15:37,910 --> 00:15:40,420 what vector I am working with. 279 00:15:40,420 --> 00:15:42,260 It's got to hold for an arbitrary A alpha. 280 00:15:42,260 --> 00:15:47,900 So the only way that can happen, this 281 00:15:47,900 --> 00:15:58,380 means that my transformation of basis vectors 282 00:15:58,380 --> 00:16:01,607 obeys a law that looks like this. 283 00:16:01,607 --> 00:16:03,190 Now, on first inspection, you're going 284 00:16:03,190 --> 00:16:04,680 to go, ah, that's exactly what we 285 00:16:04,680 --> 00:16:08,010 got for the components of the 4-vector. 286 00:16:08,010 --> 00:16:11,120 Caution-- I'm going to remind you, 287 00:16:11,120 --> 00:16:12,900 it's actually on the board over there. 288 00:16:28,770 --> 00:16:30,570 OK? 289 00:16:30,570 --> 00:16:34,700 The barred component is playing a different role 290 00:16:34,700 --> 00:16:37,680 than the barred basis vector here. 291 00:16:37,680 --> 00:16:39,330 If you want to get the barred basis 292 00:16:39,330 --> 00:16:41,640 vector from the unbarred basis vector, 293 00:16:41,640 --> 00:16:44,740 you need to work with the inverse of this matrix. 294 00:16:44,740 --> 00:16:46,240 That's what this is telling us here. 295 00:16:46,240 --> 00:16:48,180 OK? 296 00:16:48,180 --> 00:16:51,780 All that being said, if you're just working through this 297 00:16:51,780 --> 00:16:53,347 and you've got your components set up 298 00:16:53,347 --> 00:16:54,930 and you're sort of hacking through it, 299 00:16:54,930 --> 00:16:57,765 the algorithm for you to follow is actually simple. 300 00:17:07,130 --> 00:17:10,775 Really, all we're doing is lining up the indices. 301 00:17:10,775 --> 00:17:12,650 We're summing over the ones that are repeated 302 00:17:12,650 --> 00:17:14,030 and requiring that those that are 303 00:17:14,030 --> 00:17:16,400 on both the left-hand side and the right-hand side 304 00:17:16,400 --> 00:17:19,950 appear and equal one another. 305 00:17:19,950 --> 00:17:21,770 Or as an old professor of mine liked 306 00:17:21,770 --> 00:17:25,990 to say about 12 times a lecture, line up the indices. 307 00:17:30,560 --> 00:17:32,340 So that's essentially what we're doing. 308 00:17:32,340 --> 00:17:34,670 In this case, I line up-- 309 00:17:34,670 --> 00:17:36,503 if I have my Lorenz transformation 310 00:17:36,503 --> 00:17:38,670 matrix with the barred index up top and the unbarred 311 00:17:38,670 --> 00:17:39,663 down below. 312 00:17:39,663 --> 00:17:40,830 Boom, I line up the indices. 313 00:17:40,830 --> 00:17:43,080 And that tells me what the unbarred basis components 314 00:17:43,080 --> 00:17:50,400 are from the barred ones, and vice versa for the components. 315 00:17:50,400 --> 00:17:52,560 So what basically, as I just said, that tells me 316 00:17:52,560 --> 00:17:56,250 if I actually want to get this guy given this guy, 317 00:17:56,250 --> 00:17:59,590 I need to work with the inverse. 318 00:17:59,590 --> 00:18:01,840 Put this up high so that everyone in the back 319 00:18:01,840 --> 00:18:02,340 can see it. 320 00:18:08,330 --> 00:18:16,140 So this, again, is one of those places where 321 00:18:16,140 --> 00:18:20,850 you might be tempted to sort of write out a matrix 322 00:18:20,850 --> 00:18:23,490 and do a matrix inversion. 323 00:18:23,490 --> 00:18:26,870 But before you do that, remember this is physics. 324 00:18:26,870 --> 00:18:28,080 OK? 325 00:18:28,080 --> 00:18:32,040 The inverse is going to be-- the inverse matrix is going to be 326 00:18:32,040 --> 00:18:35,110 the one that does, at least for certain quantities-- 327 00:18:35,110 --> 00:18:37,110 what the Lorenz transformation does 328 00:18:37,110 --> 00:18:40,260 is that it relates objects, if I'm at rest, 329 00:18:40,260 --> 00:18:42,930 if I consider myself at rest, it tells me about things 330 00:18:42,930 --> 00:18:45,898 according to a frame that it's moving with respect to me. 331 00:18:45,898 --> 00:18:47,440 The inverse matrix does the opposite. 332 00:18:47,440 --> 00:18:48,930 It would say to that person being 333 00:18:48,930 --> 00:18:51,540 at rest, what is the matrix that tells them 334 00:18:51,540 --> 00:18:53,820 about things according to me, and they 335 00:18:53,820 --> 00:18:56,050 see me moving with the exact same velocity, 336 00:18:56,050 --> 00:18:58,360 but in the opposite direction. 337 00:18:58,360 --> 00:19:00,990 So to get the inverse matrix, to get the inverse Lorenz 338 00:19:00,990 --> 00:19:12,250 transformation, what we end up doing 339 00:19:12,250 --> 00:19:14,978 is we just reverse the velocity. 340 00:19:23,040 --> 00:19:25,120 So I've just wrote down for you-- 341 00:19:25,120 --> 00:19:27,540 and it's worth bearing in mind, every one of these lambdas 342 00:19:27,540 --> 00:19:29,910 that is there, they are really functions. 343 00:19:29,910 --> 00:19:44,510 And so my e alpha that I wrote down over there, 344 00:19:44,510 --> 00:19:48,668 I'm going to use an under tilde to denote a 3 vector. 345 00:19:48,668 --> 00:19:50,960 In one of your textbooks, it's written with a boldface. 346 00:19:50,960 --> 00:19:53,235 But that's hard to do on a blackboard. 347 00:19:53,235 --> 00:19:54,860 So if I want to go the other direction, 348 00:19:54,860 --> 00:19:57,983 well, I just need to have the inverse transformation. 349 00:20:10,820 --> 00:20:12,070 And I have that. 350 00:20:12,070 --> 00:20:15,670 Bear in mind, I mean, those are really the exact same matrices. 351 00:20:15,670 --> 00:20:16,315 OK? 352 00:20:16,315 --> 00:20:18,730 In terms of the function that I'm working with here, 353 00:20:18,730 --> 00:20:20,320 I'm just flipping around the direction in order 354 00:20:20,320 --> 00:20:21,300 to get these things out. 355 00:20:21,300 --> 00:20:21,800 OK? 356 00:20:24,260 --> 00:20:37,770 So, as I said, you might be tempted just 357 00:20:37,770 --> 00:20:42,773 to go ahead and do the matrix inversion. 358 00:20:42,773 --> 00:20:44,190 Let we just do a quick calculation 359 00:20:44,190 --> 00:20:47,333 to show you that that would work. 360 00:20:47,333 --> 00:20:48,750 And the reason I'm doing this is I 361 00:20:48,750 --> 00:20:50,917 just want to quickly step through a particular step, 362 00:20:50,917 --> 00:20:54,090 which is, again, sort of in the spirit of swatting mosquitoes 363 00:20:54,090 --> 00:20:56,280 with sledgehammers, it's the kind of analysis 364 00:20:56,280 --> 00:20:58,572 that you're going to sort of have to do off to the side 365 00:20:58,572 --> 00:21:00,060 now and again. 366 00:21:00,060 --> 00:21:05,340 So given that I know e alpha is-- 367 00:21:05,340 --> 00:21:08,060 let's see, let's use beta-- 368 00:21:08,060 --> 00:21:14,390 e beta bar be e beta bar. 369 00:21:14,390 --> 00:21:24,020 But I now know that I could write this guy as lambda-- 370 00:21:24,020 --> 00:21:25,980 let's use a gamma-- 371 00:21:25,980 --> 00:21:33,425 beta bar minus v in the gamma direction. 372 00:21:33,425 --> 00:21:35,300 Now, you look at that, and you go, ooh, look, 373 00:21:35,300 --> 00:21:38,730 I'm actually summing over the betas there. 374 00:21:38,730 --> 00:21:41,048 Let's gather my terms a little bit differently. 375 00:21:56,690 --> 00:22:01,830 So notice, I have unbarred basis vectors over here on the left, 376 00:22:01,830 --> 00:22:04,140 unbarred ones over here on the right, 377 00:22:04,140 --> 00:22:06,900 a bunch of junk in these braces here in the middle. 378 00:22:06,900 --> 00:22:10,020 The only way for that to work is if after summing over beta, 379 00:22:10,020 --> 00:22:13,030 that bunch of junk is, in matrix language, we'd say, 380 00:22:13,030 --> 00:22:14,520 it's the identity matrix. 381 00:22:14,520 --> 00:22:17,278 In component notation, we're going 382 00:22:17,278 --> 00:22:18,570 to call it the Kronecker delta. 383 00:22:48,710 --> 00:22:51,170 Just as little aside, if I started 384 00:22:51,170 --> 00:22:54,260 with the barred on the left side and the unbarred 385 00:22:54,260 --> 00:22:57,020 in the right-hand side and did a similar analysis, 386 00:22:57,020 --> 00:23:03,860 it would take you at this point now that you are all fully 387 00:23:03,860 --> 00:23:08,120 expert in this kind of index manipulation, 388 00:23:08,120 --> 00:23:10,850 it should take you no more than about a minute 389 00:23:10,850 --> 00:23:24,250 to demonstrate to yourself that you can get a Kronecker 390 00:23:24,250 --> 00:23:27,460 delta on the barred indices in a similar way, 391 00:23:27,460 --> 00:23:29,540 just by using them in a slightly different order. 392 00:23:29,540 --> 00:23:30,850 OK? 393 00:23:30,850 --> 00:23:35,290 All right, so far all still basically just formulas. 394 00:23:35,290 --> 00:23:39,980 So I'm going to start now doing a little bit of formalism that 395 00:23:39,980 --> 00:23:41,918 will very quickly segue into physics. 396 00:23:41,918 --> 00:23:43,460 We can all take a deep sigh of relief 397 00:23:43,460 --> 00:23:45,560 and go, ah, OK, something you can imagine measuring. 398 00:23:45,560 --> 00:23:46,185 So that's nice. 399 00:23:49,080 --> 00:23:53,360 So I have introduced 4-vectors. 400 00:23:53,360 --> 00:23:55,860 I haven't introduced the basis vectors and their components. 401 00:23:55,860 --> 00:23:58,800 I haven't really done anything with them yet. 402 00:23:58,800 --> 00:24:00,810 So before we start doing things with them, 403 00:24:00,810 --> 00:24:02,190 let's think about some operations 404 00:24:02,190 --> 00:24:04,560 that we can do with 4-vectors. 405 00:24:04,560 --> 00:24:07,830 So the first one which I'd like to introduce 406 00:24:07,830 --> 00:24:08,685 is a scalar product. 407 00:24:13,670 --> 00:24:15,710 To motivate the scalar product that I'm 408 00:24:15,710 --> 00:24:21,800 going to define in the same way that I defined 4-vectors 409 00:24:21,800 --> 00:24:25,490 as a quartet of numbers whose transformation properties 410 00:24:25,490 --> 00:24:29,210 are based on the transformation properties of the displacement, 411 00:24:29,210 --> 00:24:31,730 I'm going to motivate a general scalar 412 00:24:31,730 --> 00:24:37,280 product between 4-vectors by a similar kind of quantity that 413 00:24:37,280 --> 00:24:39,570 is constructed from the displacement. 414 00:24:39,570 --> 00:24:41,660 So let me recall a result that I hope 415 00:24:41,660 --> 00:24:44,074 is familiar from special relativity. 416 00:24:48,720 --> 00:25:01,750 So working in units where the speed of light is 1, 417 00:25:01,750 --> 00:25:03,610 you are hopefully all familiar with the fact 418 00:25:03,610 --> 00:25:07,150 that this quantity is something that is 419 00:25:07,150 --> 00:25:09,260 in variance to all observers. 420 00:25:09,260 --> 00:25:11,200 I did not use a represented by symbol 421 00:25:11,200 --> 00:25:13,570 there, because no matter whose delta t, delta 422 00:25:13,570 --> 00:25:15,820 x, delta y, delta z I use there, they 423 00:25:15,820 --> 00:25:21,190 will all agree on the delta s squared that comes out of this. 424 00:25:21,190 --> 00:25:22,578 If I want to-- 425 00:25:22,578 --> 00:25:24,370 actually, let me write of a few things done 426 00:25:24,370 --> 00:25:25,630 before I say anything more. 427 00:25:25,630 --> 00:25:27,430 So this is an invariant. 428 00:25:37,010 --> 00:25:37,805 This is the same-- 429 00:25:39,967 --> 00:25:42,550 let me actually write this in a slightly different way-- it is 430 00:25:42,550 --> 00:25:44,550 the same in all Lorenz frames. 431 00:25:56,960 --> 00:26:02,090 So pick some observer, get their delta t, delta x, et cetera, 432 00:26:02,090 --> 00:26:04,760 assemble delta s squared, pick a different observer, 433 00:26:04,760 --> 00:26:08,090 do the Lorenz transformation, assemble their delta t prime, 434 00:26:08,090 --> 00:26:12,350 delta x prime, et cetera, make that, boom, they all agree. 435 00:26:12,350 --> 00:26:15,290 So we're going to use this to say, 436 00:26:15,290 --> 00:26:19,340 you know what, I'm going to call that the inner product 437 00:26:19,340 --> 00:26:21,080 of the displacement vector with itself. 438 00:26:29,630 --> 00:26:37,510 So I'm going to call this delta x dotted into delta x. 439 00:26:37,510 --> 00:26:42,185 And so what this means is that built into my scalar product-- 440 00:26:44,950 --> 00:26:48,140 so if I write this as a particular observer 441 00:26:48,140 --> 00:26:49,265 would compute it-- 442 00:27:01,792 --> 00:27:03,250 this is the scalar product that I'm 443 00:27:03,250 --> 00:27:06,455 going to define with respect to the displacement vector. 444 00:27:06,455 --> 00:27:08,830 And this is usually the point where somebody in the class 445 00:27:08,830 --> 00:27:11,290 is thinking, why is there a minus sign 446 00:27:11,290 --> 00:27:12,790 in front of the timelike piece? 447 00:27:17,800 --> 00:27:19,810 I can't answer that. 448 00:27:19,810 --> 00:27:22,320 All I can say is that appears to be-- 449 00:27:22,320 --> 00:27:23,320 more than appears to be. 450 00:27:23,320 --> 00:27:26,090 There's a whole frickin' butt load of evidence-- that's 451 00:27:26,090 --> 00:27:27,817 not how nature is assembled. 452 00:27:27,817 --> 00:27:29,650 OK, it's connected to the fact-- so the fact 453 00:27:29,650 --> 00:27:31,630 that all of my spatial directions 454 00:27:31,630 --> 00:27:33,550 are sort of entered with the same sign, 455 00:27:33,550 --> 00:27:37,780 but my timelike direction, it has a different sign. 456 00:27:37,780 --> 00:27:40,060 It's connected to the fact that I can easily move 457 00:27:40,060 --> 00:27:41,863 left and right, front and back. 458 00:27:41,863 --> 00:27:43,030 It's a little bit of effort. 459 00:27:43,030 --> 00:27:44,980 I can move up and down. 460 00:27:44,980 --> 00:27:48,760 But I cannot say, oh, crap, I left my phone at home I'll go 461 00:27:48,760 --> 00:27:50,470 back 15 minutes and pick it up. 462 00:27:50,470 --> 00:27:53,170 You cannot move back and forth in time. 463 00:27:53,170 --> 00:27:55,480 Time actually, which is the timelike component 464 00:27:55,480 --> 00:27:57,515 of this thing, it enters into the geometry 465 00:27:57,515 --> 00:27:59,890 in a fundamentally different way from the spatial things. 466 00:27:59,890 --> 00:28:02,830 And that's reflected in that minus sign. 467 00:28:02,830 --> 00:28:06,580 Anything deeper than that, let's just 468 00:28:06,580 --> 00:28:09,700 say that we're probably not likely to get very 469 00:28:09,700 --> 00:28:11,560 far with that conversation. 470 00:28:11,560 --> 00:28:13,390 Depending on what kind of muscle relaxants 471 00:28:13,390 --> 00:28:14,940 you enjoy using on the weekend, you 472 00:28:14,940 --> 00:28:16,810 might have some fun conversations about it. 473 00:28:16,810 --> 00:28:19,525 But it is not something that you're really 474 00:28:19,525 --> 00:28:20,650 going to get very far with. 475 00:28:20,650 --> 00:28:21,640 You just kind of have to accept that it's 476 00:28:21,640 --> 00:28:23,470 part of the built-in geometry of nature. 477 00:28:27,040 --> 00:28:31,110 OK, so this I am defining as the inner products 478 00:28:31,110 --> 00:28:33,990 of the displacement vector with itself. 479 00:28:33,990 --> 00:28:37,350 I define vectors as having the same transformation properties 480 00:28:37,350 --> 00:28:38,550 as the displacement vector. 481 00:29:02,830 --> 00:29:06,490 We can similarly define an inner product of a 4-vector 482 00:29:06,490 --> 00:29:07,020 with itself. 483 00:29:10,275 --> 00:29:12,400 OK, now, I'll put this on another board. 484 00:29:32,310 --> 00:29:38,040 So A dot A, I'm going to define this-- 485 00:29:38,040 --> 00:29:40,530 or rather I will say it is represented according 486 00:29:40,530 --> 00:29:53,760 to observer O as minus A0 squared plus A1 squared plus A2 487 00:29:53,760 --> 00:29:59,470 squared plus A3 squared. 488 00:29:59,470 --> 00:30:01,470 I realize there can be a little bit of ambiguity 489 00:30:01,470 --> 00:30:03,200 in the way I'm writing it here. 490 00:30:03,200 --> 00:30:04,250 You just have to-- 491 00:30:04,250 --> 00:30:06,450 if you're ever confused, just ask for clarification 492 00:30:06,450 --> 00:30:08,520 about whether I'm writing something to a power 493 00:30:08,520 --> 00:30:11,070 or whether it's an index label. 494 00:30:11,070 --> 00:30:13,050 Context usually makes it clear. 495 00:30:13,050 --> 00:30:17,730 Handwriting sometimes obscures context though. 496 00:30:17,730 --> 00:30:20,880 The reason why I'm doing this and a real benefit of this 497 00:30:20,880 --> 00:30:23,130 is that whatever this quantity is, 498 00:30:23,130 --> 00:30:26,100 because A has the same transformation 499 00:30:26,100 --> 00:30:28,650 properties of the displacement, this must 500 00:30:28,650 --> 00:30:30,450 be a Lorenz invariant as well. 501 00:30:41,730 --> 00:30:44,040 The underlying mathematics of Lorenz transformation 502 00:30:44,040 --> 00:30:46,980 doesn't care that I wrote a zero instead of delta x here. 503 00:30:46,980 --> 00:30:49,980 It doesn't care that I wrote A1 instead of delta x1, et cetera. 504 00:30:49,980 --> 00:30:51,480 It just knows that it's a thing that 505 00:30:51,480 --> 00:30:53,940 goes into that slot of the Lorenz transformation. 506 00:30:53,940 --> 00:30:56,220 So all observers-- this is how I represent it 507 00:30:56,220 --> 00:31:01,325 according to observer O. But all observers agree on that form. 508 00:31:01,325 --> 00:31:03,450 And as a consequence, this is going to be something 509 00:31:03,450 --> 00:31:05,670 that we exploit a lot. 510 00:31:05,670 --> 00:31:07,830 Even when we move beyond the simple geometry 511 00:31:07,830 --> 00:31:10,230 of special relativity, a generalization of this 512 00:31:10,230 --> 00:31:15,280 will be extremely important for, not an exaggeration to say, 513 00:31:15,280 --> 00:31:17,530 everything. 514 00:31:17,530 --> 00:31:19,247 So a little bit of terminology-- 515 00:31:23,360 --> 00:31:31,210 so if A dot A is negative, and depending upon which is bigger, 516 00:31:31,210 --> 00:31:33,550 A0 squared or the sum of the other ones-- 517 00:31:33,550 --> 00:31:35,500 it could very well be negative-- 518 00:31:35,500 --> 00:31:38,590 we say that A is a time like vector. 519 00:31:43,950 --> 00:31:46,130 This traces back to the fact that if A 520 00:31:46,130 --> 00:31:49,850 were the displacement vector, if the displacement, 521 00:31:49,850 --> 00:31:52,610 the invariant interval, were negative that would tell me 522 00:31:52,610 --> 00:31:54,943 that the two events, which are the beginning and the end 523 00:31:54,943 --> 00:31:57,290 of the interval, I could find a frame at which they're 524 00:31:57,290 --> 00:32:01,033 at the exact same location and are only separated in time. 525 00:32:01,033 --> 00:32:02,450 In the same way, this is basically 526 00:32:02,450 --> 00:32:03,980 saying that I can find a frame when 527 00:32:03,980 --> 00:32:08,180 this vector points parallel to some observer's time axis. 528 00:32:13,916 --> 00:32:19,070 If this is positive, A is spacelike. 529 00:32:22,850 --> 00:32:25,430 Everything I just said about timelike, lather, rinse, 530 00:32:25,430 --> 00:32:27,903 repeat, but replace time with space. 531 00:32:27,903 --> 00:32:28,403 OK? 532 00:32:31,840 --> 00:32:37,780 And if A dot A equals 0, we say A is-- 533 00:32:37,780 --> 00:32:41,160 there's two words that are commonly used-- 534 00:32:41,160 --> 00:32:45,580 lightlike or null. 535 00:32:48,160 --> 00:32:50,200 Null just traces back obviously to the zero. 536 00:32:50,200 --> 00:32:54,280 Lightlike is because this is a vector that could lie tangent 537 00:32:54,280 --> 00:32:58,136 to the trajectory that a light beam follows in spacetime. 538 00:32:58,136 --> 00:32:58,636 OK? 539 00:33:23,780 --> 00:33:26,520 OK, let me get some clean chalk. 540 00:33:29,930 --> 00:33:32,240 So, so far, I've only talked about this inner product, 541 00:33:32,240 --> 00:33:33,680 this scalar product. 542 00:33:33,680 --> 00:33:37,340 Oh, and, by the way, I'll use inner product and scalar 543 00:33:37,340 --> 00:33:40,035 products somewhat interchangeably. 544 00:33:40,035 --> 00:33:41,660 But this allows me to reiterate a point 545 00:33:41,660 --> 00:33:43,070 I made in Tuesday's lecture. 546 00:33:43,070 --> 00:33:47,270 When I say scalar, scalar refers to a quantity which is-- 547 00:33:47,270 --> 00:33:49,770 you know, it doesn't have any components associated with it. 548 00:33:49,770 --> 00:33:52,312 So in that sense, it's familiar from your eager intuition of, 549 00:33:52,312 --> 00:33:53,758 you know, not to vector. 550 00:33:53,758 --> 00:33:55,550 But it has a deeper meaning in this course, 551 00:33:55,550 --> 00:33:56,540 because I also want it to be something 552 00:33:56,540 --> 00:33:58,910 that is invariant between reference frames. 553 00:33:58,910 --> 00:34:01,100 So A dot A is the scalar product. 554 00:34:01,100 --> 00:34:04,280 It gives me a quantity that all observers agree on. 555 00:34:04,280 --> 00:34:06,860 Now, I've only done scalar products 556 00:34:06,860 --> 00:34:09,020 of vectors, A and the displacement vector, 557 00:34:09,020 --> 00:34:10,460 with themselves. 558 00:34:10,460 --> 00:34:22,949 So a more general notion, if I have vectors A and B, 559 00:34:22,949 --> 00:34:25,409 then I will define the inner product between them, 560 00:34:25,409 --> 00:34:30,510 as observed by O, as constructed by observer O, rather, like so. 561 00:34:43,190 --> 00:34:44,732 It's not hard to convince yourself, 562 00:34:44,732 --> 00:34:46,190 given everything we've done so far, 563 00:34:46,190 --> 00:34:49,605 that this quantity must also be invariant. 564 00:34:49,605 --> 00:34:50,980 I'll sketch a really quick proof. 565 00:34:58,570 --> 00:35:01,060 Let's define-- let's say we have two 4-vectors, A 566 00:35:01,060 --> 00:35:06,760 and B. Their sum, by the linearity rules that 567 00:35:06,760 --> 00:35:09,530 apply to these vectors, must also be a vector. 568 00:35:09,530 --> 00:35:17,620 And so if I compute, C dot C, this is an invariant. 569 00:35:17,620 --> 00:35:27,340 With a little bit of labor, that basically 570 00:35:27,340 --> 00:35:31,100 boils down to middle school algebra. 571 00:35:31,100 --> 00:35:33,570 You can show that C dot C is A dot A plus B dot B 572 00:35:33,570 --> 00:35:37,270 plus twice A dot B. This is invariant. 573 00:35:37,270 --> 00:35:39,070 This is invariant. 574 00:35:39,070 --> 00:35:40,000 This is invariant. 575 00:35:40,000 --> 00:35:42,760 And so this must be invariant. 576 00:35:49,010 --> 00:35:53,068 So this is really useful for us, because we now have a way-- 577 00:35:53,068 --> 00:35:55,360 I've introduced these objects, these geometric objects, 578 00:35:55,360 --> 00:35:56,480 these 4-vectors. 579 00:35:56,480 --> 00:35:59,760 We are going to use them in this class 580 00:35:59,760 --> 00:36:01,920 to describe quantities that are of interest 581 00:36:01,920 --> 00:36:05,040 to the physicist who wants to make measurements in spacetime. 582 00:36:05,040 --> 00:36:06,498 We've now learned one of the things 583 00:36:06,498 --> 00:36:08,040 when you're doing stuff in relativity 584 00:36:08,040 --> 00:36:10,270 is you have to be careful who is measuring what. 585 00:36:10,270 --> 00:36:12,630 What are the components that 4-vector as seen 586 00:36:12,630 --> 00:36:14,038 by this observer? 587 00:36:14,038 --> 00:36:16,580 What about their friend who's jogging through the room at 3/4 588 00:36:16,580 --> 00:36:17,340 quarters speed of light? 589 00:36:17,340 --> 00:36:19,243 What about their friend who's driving 2/3 the speed of light 590 00:36:19,243 --> 00:36:20,010 in the other direction? 591 00:36:20,010 --> 00:36:22,343 You have all these really annoying calculations that you 592 00:36:22,343 --> 00:36:24,660 can and sometimes have to do. 593 00:36:24,660 --> 00:36:26,820 This gives us a way to get certain things that 594 00:36:26,820 --> 00:36:28,350 are invariant out of the situation 595 00:36:28,350 --> 00:36:30,830 that everyone is going to agree on. 596 00:36:30,830 --> 00:36:32,250 Invariants are our friends. 597 00:36:42,360 --> 00:36:44,860 So earlier today, earlier in today's lecture, 598 00:36:44,860 --> 00:36:48,480 I talked about how I can write my 4-vectors using the basis 599 00:36:48,480 --> 00:36:49,500 factors. 600 00:36:49,500 --> 00:36:51,410 So another way of writing this-- 601 00:36:51,410 --> 00:36:54,390 so what's sort of annoying is every time I've actually 602 00:36:54,390 --> 00:36:56,970 written out the inner product, I have used the represented 603 00:36:56,970 --> 00:36:58,890 by symbol. 604 00:36:58,890 --> 00:36:59,640 I don't want that. 605 00:36:59,640 --> 00:37:01,900 I want to have equal symbols in there, dammit. 606 00:37:01,900 --> 00:37:09,390 So let's take advantage of the fact that A dot B, 607 00:37:09,390 --> 00:37:13,240 I know how to expand A and B using components and basis 608 00:37:13,240 --> 00:37:13,740 vectors. 609 00:37:25,770 --> 00:37:27,790 And again, using the index notation, 610 00:37:27,790 --> 00:37:29,670 I can just pull everything out and rearrange 611 00:37:29,670 --> 00:37:30,420 this a little bit. 612 00:37:43,588 --> 00:37:45,380 Whenever you get down to a point like this, 613 00:37:45,380 --> 00:37:48,500 we now get to do what every mathematician loves to do-- 614 00:37:48,500 --> 00:37:50,360 give something a name. 615 00:37:50,360 --> 00:37:53,750 I'm going to define the inner product of basis vector 616 00:37:53,750 --> 00:38:01,800 A with basis vector B to be a two index tensor-- 617 00:38:01,800 --> 00:38:03,840 eta alpha beta. 618 00:38:06,910 --> 00:38:12,430 What's lovely about this, this is a totally frame invariant 619 00:38:12,430 --> 00:38:12,940 quantity. 620 00:38:12,940 --> 00:38:14,060 We know that. 621 00:38:14,060 --> 00:38:18,060 And so I've now found a way to write this using the components 622 00:38:18,060 --> 00:38:20,350 as something that gives me a result that is totally 623 00:38:20,350 --> 00:38:21,435 frame invariant. 624 00:38:21,435 --> 00:38:23,060 Now, when you hack through a little bit 625 00:38:23,060 --> 00:38:24,820 the algebra of this, what you'll find 626 00:38:24,820 --> 00:38:27,220 is that the components of this metric-- 627 00:38:27,220 --> 00:38:29,610 oh, shoot, I didn't want to actually say it out loud-- 628 00:38:29,610 --> 00:38:31,802 the components of this tensor, which pretend 629 00:38:31,802 --> 00:38:33,010 you didn't hear me say that-- 630 00:38:33,010 --> 00:38:37,208 the components of this tensor has the following components-- 631 00:38:37,208 --> 00:38:39,625 I just said something circular-- has the following values. 632 00:38:49,250 --> 00:38:52,130 This is, as I unfortunately gave away 633 00:38:52,130 --> 00:38:55,010 the plot, this is, in fact, the metric 634 00:38:55,010 --> 00:38:58,250 that I said at the beginning is the quantity that I must 635 00:38:58,250 --> 00:38:59,810 associate with spacetime in order 636 00:38:59,810 --> 00:39:02,210 for there to be a notion of distance between events. 637 00:39:12,510 --> 00:39:15,137 I haven't really said what a tensor is carefully yet. 638 00:39:15,137 --> 00:39:17,220 I'm going to make a more formal definition of this 639 00:39:17,220 --> 00:39:18,040 in just a moment. 640 00:39:18,040 --> 00:39:19,623 But this is your first example of one. 641 00:39:23,150 --> 00:39:26,020 And so the way in which this actually 642 00:39:26,020 --> 00:39:29,170 gives me a notion of distance is through this 643 00:39:29,170 --> 00:39:31,360 that I wrote down right here. 644 00:39:31,360 --> 00:39:37,840 If I have two events in spacetime 645 00:39:37,840 --> 00:39:49,410 that are separated by a displacement delta x, 646 00:39:49,410 --> 00:39:55,770 the delta s squared, which I obtained from this thing, 647 00:39:55,770 --> 00:39:59,160 is fundamentally the notion of distance between those two 648 00:39:59,160 --> 00:40:00,420 events that I use. 649 00:40:00,420 --> 00:40:03,450 And notice, it's a little less normal of a distance 650 00:40:03,450 --> 00:40:04,950 than you're used to when you do sort 651 00:40:04,950 --> 00:40:06,400 of ordinary Euclidean geometry. 652 00:40:06,400 --> 00:40:09,786 This is a distance whose square can be negative. 653 00:40:20,480 --> 00:40:22,370 What we like to say is that when you're 654 00:40:22,370 --> 00:40:26,960 working in special relativity, it's 655 00:40:26,960 --> 00:40:29,480 not necessarily positive-- the distance between two events 656 00:40:29,480 --> 00:40:31,480 is not necessarily-- the distance squared is not 657 00:40:31,480 --> 00:40:34,162 necessarily positive definite. 658 00:40:34,162 --> 00:40:35,870 If it's negative, though, that just means 659 00:40:35,870 --> 00:40:38,390 it's sort of dominated by the time interval between them. 660 00:40:38,390 --> 00:40:40,640 If it's positive, you know it's dominated by the space 661 00:40:40,640 --> 00:40:41,900 interval between them. 662 00:40:41,900 --> 00:40:44,375 If it's zero, well, you actually know-- 663 00:40:44,375 --> 00:40:46,500 it's actually a little bit confusing at that point. 664 00:40:46,500 --> 00:40:48,490 They could be, in fact, you know, 665 00:40:48,490 --> 00:40:51,380 very widely separated in both space and time, 666 00:40:51,380 --> 00:40:54,080 but in such a way that a light beam could connect them. 667 00:40:54,080 --> 00:40:57,940 So there's a lot of information encoded in that. 668 00:40:57,940 --> 00:41:00,832 Now, as we move forward-- 669 00:41:00,832 --> 00:41:02,540 hang on a second-- 670 00:41:02,540 --> 00:41:08,020 as we move forward, we're going to upgrade this. 671 00:41:08,020 --> 00:41:12,760 So right now, our metric is just this simple matrix 672 00:41:12,760 --> 00:41:17,160 of minus 1s, 0s, and 1s. 673 00:41:17,160 --> 00:41:19,180 One of things that we're going to do 674 00:41:19,180 --> 00:41:22,700 is sort of a warm-up exercise to the more complicated things 675 00:41:22,700 --> 00:41:24,750 we're going to do later is we're going 676 00:41:24,750 --> 00:41:26,737 to move away from special relativity 677 00:41:26,737 --> 00:41:27,820 and Cartesian coordinates. 678 00:41:27,820 --> 00:41:29,160 We're going to look at it in polar coordinates. 679 00:41:29,160 --> 00:41:31,118 That's going to be kind of like a warm-up zone. 680 00:41:31,118 --> 00:41:33,790 And so when we do that, we're always 681 00:41:33,790 --> 00:41:38,740 going to reserve eta for the metric of special relativity 682 00:41:38,740 --> 00:41:40,920 when I'm working in Cartesian coordinates. 683 00:41:40,920 --> 00:41:42,670 It's just a great symbol to have for that. 684 00:41:42,670 --> 00:41:44,320 And it's a useful thing to always 685 00:41:44,320 --> 00:41:47,130 have that definition in mind. 686 00:41:47,130 --> 00:41:49,030 I can continue to do special relativity, 687 00:41:49,030 --> 00:41:50,710 but then working in coordinates that are, you know, 688 00:41:50,710 --> 00:41:53,050 spherical-like or polar-like or something like that, 689 00:41:53,050 --> 00:41:55,960 then this is going to become a function. 690 00:41:55,960 --> 00:41:58,000 And what that is going to mean is 691 00:41:58,000 --> 00:41:59,680 that things like my little basis vector 692 00:41:59,680 --> 00:42:02,580 is going to have more complicated behavior. 693 00:42:02,580 --> 00:42:04,330 A little later in the course, we will then 694 00:42:04,330 --> 00:42:08,380 show that when gravity enters into the picture, 695 00:42:08,380 --> 00:42:09,910 essentially the essence of gravity 696 00:42:09,910 --> 00:42:12,670 is going to be encoded in this thing as well in a way 697 00:42:12,670 --> 00:42:14,420 where, again, it's going to be a function. 698 00:42:14,420 --> 00:42:16,810 It's going to vary as a function of space and time. 699 00:42:16,810 --> 00:42:19,175 And the dynamics of gravity will be buried in that. 700 00:42:19,175 --> 00:42:21,550 It's sort of funny that it really does just sort of start 701 00:42:21,550 --> 00:42:22,050 out-- 702 00:42:22,050 --> 00:42:24,940 I mean, if you take that thing and you set delta t equals 0, 703 00:42:24,940 --> 00:42:27,100 this is just the bloody Pythagorean theorem. 704 00:42:27,100 --> 00:42:30,820 That is all this is. 705 00:42:30,820 --> 00:42:33,430 Put time back in, and it sort of is the generalization 706 00:42:33,430 --> 00:42:35,055 of Pythagoras to spacetime. 707 00:42:35,055 --> 00:42:37,180 And, in fact, we're going to take advantage of that 708 00:42:37,180 --> 00:42:39,610 and sort of define a geometry that 709 00:42:39,610 --> 00:42:41,698 looks like this as being flat in the same way 710 00:42:41,698 --> 00:42:43,990 that a board is flat, and the Pythagorean theorem works 711 00:42:43,990 --> 00:42:45,258 perfectly on it. 712 00:42:45,258 --> 00:42:47,050 Then, we're going to start think about what 713 00:42:47,050 --> 00:42:48,370 happens when it becomes curved. 714 00:42:48,370 --> 00:42:49,420 And you start thinking about things 715 00:42:49,420 --> 00:42:51,920 like, what is the geometry on the surface of the sphere look 716 00:42:51,920 --> 00:42:52,432 like? 717 00:42:52,432 --> 00:42:53,890 That's just sort of pointing ahead. 718 00:42:53,890 --> 00:42:55,557 So I just throw that at you, so that you 719 00:42:55,557 --> 00:42:57,520 get ready for some of the concepts 720 00:42:57,520 --> 00:43:01,090 that we'll be talking about soon. 721 00:43:01,090 --> 00:43:04,070 So let me write this actually in terms of differentials. 722 00:43:04,070 --> 00:43:06,070 It's sort of useful for what I want to say next. 723 00:43:12,380 --> 00:43:16,180 So a little differential, if I have two events in spacetime 724 00:43:16,180 --> 00:43:22,670 that are very close to one another, 725 00:43:22,670 --> 00:43:25,140 I can write them like so. 726 00:43:25,140 --> 00:43:26,630 And what I've essentially done here 727 00:43:26,630 --> 00:43:35,870 is written dx as dx alpha e alpha. 728 00:43:35,870 --> 00:43:37,762 Before I get into some more sort of a couple 729 00:43:37,762 --> 00:43:39,470 of important, fairly important 4-vectors, 730 00:43:39,470 --> 00:43:42,860 the reason I did this is I want to make an important point 731 00:43:42,860 --> 00:43:46,780 about some notation and terminology that is used. 732 00:43:49,380 --> 00:43:54,570 If it is the case that the displacement vector is related 733 00:43:54,570 --> 00:44:07,310 to the differentials of your coordinates like so, 734 00:44:07,310 --> 00:44:18,590 we say that e alpha is a coordinate basis vector. 735 00:44:24,490 --> 00:44:31,350 What it does is it transforms a differential of your coordinate 736 00:44:31,350 --> 00:44:37,782 into a differential vector in spacetime. 737 00:44:37,782 --> 00:44:39,990 Now, you, may be thinking to yourself, OK, well, what 738 00:44:39,990 --> 00:44:41,060 other kind can there be? 739 00:44:49,780 --> 00:44:51,660 Well, this is where my little spiel there 740 00:44:51,660 --> 00:44:53,077 a second ago about how we're going 741 00:44:53,077 --> 00:44:56,155 to start looking at more complicated things, 742 00:44:56,155 --> 00:44:57,460 it's going to become important. 743 00:44:57,460 --> 00:44:59,790 So when we're working in a Cartesian-like coordinate 744 00:44:59,790 --> 00:45:03,150 system, the fact that this is what we call a coordinate basis 745 00:45:03,150 --> 00:45:05,743 vector isn't very interesting. 746 00:45:26,945 --> 00:45:29,070 Suppose I was working in some kind of a curvilinear 747 00:45:29,070 --> 00:45:30,360 coordinate system, OK? 748 00:45:30,360 --> 00:45:32,010 Spherical coordinates. 749 00:45:32,010 --> 00:45:34,290 Now, let's just focus on 3-space for a second. 750 00:45:47,850 --> 00:45:52,320 So if I write a sort of analogous equation 751 00:45:52,320 --> 00:45:57,820 in curvilinear coordinates-- 752 00:45:57,820 --> 00:46:00,285 OK, so here's the 3-space version of that. 753 00:46:00,285 --> 00:46:04,020 Now let's imagine that i equals 1 corresponds to radius, 754 00:46:04,020 --> 00:46:06,660 i equals 2 is theta, i equals 3 is phi. 755 00:46:09,360 --> 00:46:22,943 Then, this would be dr er plus d theta e theta plus d phi e phi. 756 00:46:27,380 --> 00:46:30,560 Does that disturb you at all? 757 00:46:30,560 --> 00:46:35,120 OK, this has dimensions of length. 758 00:46:35,120 --> 00:46:39,690 These have the dimensions of angle. 759 00:46:39,690 --> 00:46:44,340 In order for this to work, er must be dimensionless. 760 00:46:44,340 --> 00:46:46,730 e theta must have the dimensions of length. 761 00:46:46,730 --> 00:46:48,690 e phi must have the dimensions of length. 762 00:46:51,370 --> 00:46:54,640 This is what a coordinate basis looks 763 00:46:54,640 --> 00:46:57,130 like when I am dealing with-- 764 00:46:57,130 --> 00:46:59,660 well, we're going to use is a lot in this thing. 765 00:46:59,660 --> 00:47:03,070 I introduce this right now because you are all probably 766 00:47:03,070 --> 00:47:04,870 looking at that, and some small part of you 767 00:47:04,870 --> 00:47:08,140 inside is weeping, because what you want me to write down 768 00:47:08,140 --> 00:47:08,695 is this. 769 00:47:20,975 --> 00:47:23,200 Ah, isn't that better? 770 00:47:23,200 --> 00:47:25,380 OK, this looks like something you're used to. 771 00:47:31,260 --> 00:47:32,925 So I throw this out here right now 772 00:47:32,925 --> 00:47:34,550 just because I want to make sure you're 773 00:47:34,550 --> 00:47:38,030 aware that there are some equations 774 00:47:38,030 --> 00:47:40,408 and some foundational stuff you guys have been doing 775 00:47:40,408 --> 00:47:42,950 over the years, particularly, this shows up a lot when you've 776 00:47:42,950 --> 00:47:46,580 done E&M out of a textbook like Purcell or Griffiths 777 00:47:46,580 --> 00:47:52,850 or Jackson, because there's some derivative operators, which 778 00:47:52,850 --> 00:47:56,560 are assuming that your basis vectors are what we call 779 00:47:56,560 --> 00:47:57,667 orthonormal. 780 00:48:05,130 --> 00:48:11,365 So my e i hat here, it is an orthonormal basis. 781 00:48:17,810 --> 00:48:20,690 And orthonormal basis is defined such 782 00:48:20,690 --> 00:48:32,170 that the dot product of any two members of this thing 783 00:48:32,170 --> 00:48:34,930 gives me back the chronic or delta. 784 00:48:34,930 --> 00:48:38,620 That is not necessarily the case when 785 00:48:38,620 --> 00:48:40,375 I work with a coordinate basis. 786 00:48:46,720 --> 00:48:52,515 Our basis has er dot er equals 1. 787 00:48:52,515 --> 00:48:54,100 Yay, that one's nice. 788 00:48:54,100 --> 00:49:02,180 But when I do e theta dot e theta, I get r squared. 789 00:49:11,680 --> 00:49:15,888 e phi dot e phi will be r squared sine squared theta. 790 00:49:15,888 --> 00:49:17,860 And what I'm going to do when I start 791 00:49:17,860 --> 00:49:20,920 generalizing these things, I'm going to change my-- 792 00:49:20,920 --> 00:49:22,497 this thing which I defined up here-- 793 00:49:22,497 --> 00:49:23,830 do I still have it on the board? 794 00:49:23,830 --> 00:49:24,788 Yeah, yeah, right here. 795 00:49:24,788 --> 00:49:29,350 So when I set eta alpha beta is e alpha dot e beta and I made 796 00:49:29,350 --> 00:49:31,660 it this thing, I'm going to generalize this and say 797 00:49:31,660 --> 00:49:34,150 at the dot product of any two basis 798 00:49:34,150 --> 00:49:37,090 vectors it gives me a more general notion 799 00:49:37,090 --> 00:49:38,680 of a metric tensor. 800 00:49:38,680 --> 00:49:40,660 And the values in the metric tensor 801 00:49:40,660 --> 00:49:43,620 maybe functions like this. 802 00:49:43,620 --> 00:49:45,370 Right now, throw that out there, you know, 803 00:49:45,370 --> 00:49:49,180 this might be sort of just like a peak of the horrors 804 00:49:49,180 --> 00:49:49,810 that lie ahead. 805 00:49:49,810 --> 00:49:52,227 OK, we're not going to worry about this too much just yet. 806 00:49:52,227 --> 00:49:53,980 But I want you to be prepared for this. 807 00:49:53,980 --> 00:49:55,570 In particular, it's really useful 808 00:49:55,570 --> 00:49:58,780 to have this notion of a coordinate basis 809 00:49:58,780 --> 00:50:01,810 versus an orthonormal basis in your head. 810 00:50:01,810 --> 00:50:04,150 We're going to start defining some derivative operations 811 00:50:04,150 --> 00:50:04,740 soon. 812 00:50:04,740 --> 00:50:06,490 In fact, probably won't get to them today, 813 00:50:06,490 --> 00:50:11,227 but they will be present when we start doing Lecture 3. 814 00:50:11,227 --> 00:50:13,060 And there's a couple of results that come up 815 00:50:13,060 --> 00:50:14,643 where everyone's sort of like, wait, I 816 00:50:14,643 --> 00:50:17,300 knew that the divergence had a factor of r on that derivative 817 00:50:17,300 --> 00:50:17,800 there. 818 00:50:17,800 --> 00:50:18,760 Where'd it go? 819 00:50:18,760 --> 00:50:23,850 It's because we're not working in an orthonormal basis. 820 00:50:23,850 --> 00:50:27,222 All right, I'm a little sick of math. 821 00:50:27,222 --> 00:50:28,430 So let's do a little physics. 822 00:50:33,470 --> 00:50:38,800 So, so far, the actual only physical 4-vector 823 00:50:38,800 --> 00:50:41,950 that I've introduced is the displacement vector. 824 00:50:57,470 --> 00:51:00,020 From the displacement vector, it's really easy 825 00:51:00,020 --> 00:51:02,180 to make the probably the first and simplest 826 00:51:02,180 --> 00:51:10,270 important 4-vector, which is known as the 4-velocity. 827 00:51:14,570 --> 00:51:18,140 This tells me the rate of displacement of an observer 828 00:51:18,140 --> 00:51:22,032 as this person moves through spacetime per unit-- 829 00:51:22,032 --> 00:51:24,365 and we're going to be careful about this in this class-- 830 00:51:26,870 --> 00:51:39,410 d tau is the time interval as measured 831 00:51:39,410 --> 00:51:53,400 along the trajectory of the observer with 4-velocity u. 832 00:51:53,400 --> 00:51:56,370 In other words-- that's a very long winded way of saying it-- 833 00:51:56,370 --> 00:51:58,440 it's an interval of proper time. 834 00:52:09,840 --> 00:52:12,323 In English, the word proper time sounds very like, woo, 835 00:52:12,323 --> 00:52:13,740 I don't want to use improper time. 836 00:52:13,740 --> 00:52:14,572 I better use that. 837 00:52:14,572 --> 00:52:16,530 But this actually I think it comes from French. 838 00:52:16,530 --> 00:52:19,680 It just refers to the fact that it's one's own time. 839 00:52:19,680 --> 00:52:21,780 Apparently in German people say eigenzeit. 840 00:52:21,780 --> 00:52:24,420 So, you know, there's a couple of different words for it. 841 00:52:24,420 --> 00:52:27,703 But proper time is what we use. 842 00:52:27,703 --> 00:52:29,370 In special relativity, if we see someone 843 00:52:29,370 --> 00:52:35,410 going by with constant velocity, a particular observer who sees, 844 00:52:35,410 --> 00:52:36,020 you know-- 845 00:52:36,020 --> 00:52:36,770 we're here in the room. 846 00:52:36,770 --> 00:52:37,687 Someone comes through. 847 00:52:37,687 --> 00:52:39,170 Their 4-velocity is u. 848 00:52:39,170 --> 00:52:42,350 We would see their 4-velocity to have the components 849 00:52:42,350 --> 00:52:48,290 gamma gamma v, where gamma, I'll remind you, 850 00:52:48,290 --> 00:52:52,750 is the special relativistic Lorenz factor. 851 00:52:52,750 --> 00:52:57,200 And I'll remind you again we've set speed of light to 1. 852 00:52:57,200 --> 00:53:00,220 A very useful thing, which we're actually 853 00:53:00,220 --> 00:53:03,760 going to take advantage of quite a bit, 854 00:53:03,760 --> 00:53:07,750 is that in the rest frame of u-- 855 00:53:24,795 --> 00:53:25,920 pardon me for a second-- 856 00:53:28,550 --> 00:53:34,200 in the rest frame of u, or I should 857 00:53:34,200 --> 00:53:47,922 say of the observer whose 4-velocity is u, 858 00:53:47,922 --> 00:53:51,340 they just have 1 in atomic direction, C, 859 00:53:51,340 --> 00:53:53,760 if you want to put your factors back into their. 860 00:53:53,760 --> 00:53:56,260 And that's basically just saying that the person is standing 861 00:53:56,260 --> 00:54:01,000 still, but moving through time, because you are always 862 00:54:01,000 --> 00:54:03,160 moving through time. 863 00:54:03,160 --> 00:54:10,600 All right, from the 4-velocity for an observer who has-- 864 00:54:10,600 --> 00:54:13,870 or for an object, I should say, who has some mass, 865 00:54:13,870 --> 00:54:24,160 we can easily define the 4-momentum, 866 00:54:24,160 --> 00:54:28,540 where this m is known as the rest mass of this object. 867 00:54:36,921 --> 00:54:40,820 It's worth a bit of description here. 868 00:54:40,820 --> 00:54:43,430 You will often see, particularly in some older textbooks that 869 00:54:43,430 --> 00:54:44,860 discuss special relativity, people 870 00:54:44,860 --> 00:54:48,330 like to talk about the relativistic mass. 871 00:54:48,330 --> 00:54:50,480 And that comes from the fact that if I write out 872 00:54:50,480 --> 00:54:52,640 what this thing looks like according 873 00:54:52,640 --> 00:55:02,540 to some particular observer, you have this gamma m entering 874 00:55:02,540 --> 00:55:03,890 into both of the components. 875 00:55:03,890 --> 00:55:05,660 And so older textbooks often called 876 00:55:05,660 --> 00:55:08,390 gamma m the relativistic mass. 877 00:55:08,390 --> 00:55:10,348 That's not really the way people have-- 878 00:55:10,348 --> 00:55:12,140 over the course of the past couple decades, 879 00:55:12,140 --> 00:55:14,900 they've moved away from that. 880 00:55:14,900 --> 00:55:17,930 And it's just more useful to focus on the rest mass 881 00:55:17,930 --> 00:55:19,703 as the only really meaningful mass, 882 00:55:19,703 --> 00:55:21,120 because, as we'll see in a moment, 883 00:55:21,120 --> 00:55:23,180 it's a Lorenz invariant. 884 00:55:23,180 --> 00:55:27,207 We'll see how that is in literally about 3 minutes. 885 00:55:27,207 --> 00:55:28,790 And so what we're instead going to say 886 00:55:28,790 --> 00:55:33,360 is that as seen by some particular observer, 887 00:55:33,360 --> 00:55:35,100 this has a timelike component that 888 00:55:35,100 --> 00:55:38,490 is the energy that that observer would measure 889 00:55:38,490 --> 00:55:40,470 and a set of spacelike components 890 00:55:40,470 --> 00:55:42,720 that are the momentum that that observer will measure. 891 00:55:45,890 --> 00:55:48,470 So where we get a bit of important physics out of all 892 00:55:48,470 --> 00:55:53,128 this stuff is by coupling these two 4-vectors to the scalar 893 00:55:53,128 --> 00:55:54,170 products that we made up. 894 00:56:07,210 --> 00:56:12,550 So the first one, if you do u dot u, 895 00:56:12,550 --> 00:56:14,110 according to any observer, that's 896 00:56:14,110 --> 00:56:18,340 just going to be minus gamma squared 897 00:56:18,340 --> 00:56:21,300 plus gamma squared v squared. 898 00:56:25,690 --> 00:56:28,390 And with about 20 seconds worth of analysis, 899 00:56:28,390 --> 00:56:33,826 you can find that this is always equal to minus 1. 900 00:56:38,020 --> 00:56:41,390 Actually, there's an even trickier way to do this. 901 00:56:41,390 --> 00:56:44,980 Suppose I evaluate this in the rest frame of the observer 902 00:56:44,980 --> 00:56:46,690 whose 4-velocity is u? 903 00:56:46,690 --> 00:56:50,380 Well, in the rest frame, v is 0, and gamma is 1, 904 00:56:50,380 --> 00:56:51,800 and I get minus 1. 905 00:56:51,800 --> 00:56:53,080 And this is an invariant. 906 00:56:53,080 --> 00:56:55,120 So whatever I get in that particular frame 907 00:56:55,120 --> 00:56:57,285 must be obtained in all frames. 908 00:56:57,285 --> 00:56:58,660 That's a trick we're going to use 909 00:56:58,660 --> 00:56:59,868 over and over and over again. 910 00:56:59,868 --> 00:57:01,645 Sometimes you can identify-- 911 00:57:01,645 --> 00:57:03,820 you know, you get some kind of God awful expression 912 00:57:03,820 --> 00:57:04,960 that just makes you want to vomit. 913 00:57:04,960 --> 00:57:06,460 But then you go, wait a minute, what 914 00:57:06,460 --> 00:57:08,667 would this look like in frame blah, blah, blah? 915 00:57:08,667 --> 00:57:10,750 And you sort of think about some particular frame. 916 00:57:10,750 --> 00:57:13,780 And in that frame, it may simplify. 917 00:57:13,780 --> 00:57:17,890 And if it does and it's a frame invariant quantity, mazel tov, 918 00:57:17,890 --> 00:57:19,960 you have just basically won the lottery. 919 00:57:19,960 --> 00:57:22,360 You've got this all taken care of. 920 00:57:22,360 --> 00:57:24,610 Go on with your life. 921 00:57:24,610 --> 00:57:29,120 So the 4-velocity has a scalar product of itself 922 00:57:29,120 --> 00:57:30,510 that is always minus 1. 923 00:57:30,510 --> 00:57:31,010 OK? 924 00:57:37,365 --> 00:57:40,810 How about the 4-momentum? 925 00:57:40,810 --> 00:57:45,370 Well, the 4-momentum is just 4-velocity times mass. 926 00:58:07,600 --> 00:58:10,240 So that's just minus m squared. 927 00:58:10,240 --> 00:58:12,870 But we also know it's related to these two other quantities, 928 00:58:12,870 --> 00:58:14,430 which are important in physics. 929 00:58:14,430 --> 00:58:17,260 This is related to the energy and to the momentum. 930 00:58:17,260 --> 00:58:20,760 So this is also equal to minus e squared plus-- 931 00:58:24,860 --> 00:58:28,340 so this is the ordinary 3-- 932 00:58:28,340 --> 00:58:31,010 the magnitude of the 3-vector part of this thing as 933 00:58:31,010 --> 00:58:34,760 measured by the observer who breaks up 934 00:58:34,760 --> 00:58:37,040 the 4-momentum in this way. 935 00:58:37,040 --> 00:58:39,800 So what this means is I can manipulate this guy 936 00:58:39,800 --> 00:58:40,970 around a little bit here. 937 00:58:52,710 --> 00:58:55,060 Anybody who works in particle physics 938 00:58:55,060 --> 00:58:56,810 is presumably familiar with this equation. 939 00:58:56,810 --> 00:58:58,435 Sometimes it appears with the p squared 940 00:58:58,435 --> 00:59:00,160 moved onto the other side. 941 00:59:00,160 --> 00:59:02,650 If it looks a little bit unfamiliar to you, 942 00:59:02,650 --> 00:59:06,690 let me put some factors of C back in this. 943 00:59:06,690 --> 00:59:08,810 So remember, we have set C equal to 1. 944 00:59:17,560 --> 00:59:20,860 When you put it back in, that's what this is. 945 00:59:20,860 --> 00:59:23,320 So it drops out of this in a very, very simple way. 946 00:59:26,330 --> 00:59:28,840 One of the uses of this-- 947 00:59:28,840 --> 00:59:31,450 and many of you have done exercises presumably 948 00:59:31,450 --> 00:59:33,200 in some previous study that does this. 949 00:59:33,200 --> 00:59:38,277 And there'll be one exercise on the p set 950 00:59:38,277 --> 00:59:40,110 that was just posted where you exploit this. 951 00:59:43,620 --> 00:59:45,270 So a key bit of physics, the reason 952 00:59:45,270 --> 00:59:50,430 why we care about 4-momentum is it's in one mathematical object 953 00:59:50,430 --> 00:59:52,440 allows us to combine conservation energy 954 00:59:52,440 --> 00:59:53,730 and conservation of momentum. 955 01:00:01,460 --> 01:00:06,617 So conservation of 4-momentum puts both conservation 956 01:00:06,617 --> 01:00:08,200 of energy and conservation of momentum 957 01:00:08,200 --> 01:00:10,340 into one mathematical object. 958 01:00:10,340 --> 01:00:20,900 So if I have n particles that interact, 959 01:00:20,900 --> 01:00:39,143 then the total 4-momentum is conserved in the interaction. 960 01:00:45,765 --> 01:00:46,720 OK? 961 01:00:46,720 --> 01:00:50,637 So, yeah, we talk a little bit more about this 962 01:00:50,637 --> 01:00:52,220 and then just sort of quickly move on. 963 01:00:52,220 --> 01:00:56,370 So combining this with the fact that we 964 01:00:56,370 --> 01:00:59,702 are free to change our reference frames often 965 01:00:59,702 --> 01:01:01,910 it gives us a trick that allows us to really simplify 966 01:01:01,910 --> 01:01:03,190 a lot of analysis. 967 01:01:07,420 --> 01:01:09,005 So if I have n particles that are 968 01:01:09,005 --> 01:01:10,380 sort of swarming around and doing 969 01:01:10,380 --> 01:01:16,470 some horrible bit of business that I need to study 970 01:01:16,470 --> 01:01:22,020 and I need to have a good understanding of, 971 01:01:22,020 --> 01:01:37,780 we can often vastly simplify our algebra 972 01:01:37,780 --> 01:01:41,372 by choosing a special and very convenient frame of reference 973 01:01:41,372 --> 01:01:42,580 in which to do our analysis-- 974 01:01:46,340 --> 01:01:48,270 choose the center of momentum frame. 975 01:01:54,920 --> 01:02:00,255 So this is the time frame in which that that p tote-- 976 01:02:04,190 --> 01:02:06,786 so C-O-M, center of momentum-- 977 01:02:09,910 --> 01:02:12,060 has zero spatial momentum. 978 01:02:12,060 --> 01:02:14,830 In that frame, you have just as much momentum going to the left 979 01:02:14,830 --> 01:02:16,872 as going to the right, just as much going forward 980 01:02:16,872 --> 01:02:19,300 as going backwards, as much going up as going down. 981 01:02:19,300 --> 01:02:22,930 And so this turns out to be-- so the classic example of where 982 01:02:22,930 --> 01:02:29,592 this is really useful is when you are studying particle 983 01:02:29,592 --> 01:02:32,050 collisions and you're looking at things like the production 984 01:02:32,050 --> 01:02:33,253 of new particles. 985 01:02:44,860 --> 01:02:50,660 So imagine you've got particle A with some 4-momentum PA coming 986 01:02:50,660 --> 01:02:52,130 in like this. 987 01:02:52,130 --> 01:02:56,030 Particle B's got some 4-momentum coming in like this. 988 01:02:56,030 --> 01:02:58,310 These guys collide. 989 01:02:58,310 --> 01:02:59,270 And they do so-- 990 01:02:59,270 --> 01:03:00,950 I work in the center of momentum frame. 991 01:03:00,950 --> 01:03:04,490 I might want to just calculate the energy at which they just 992 01:03:04,490 --> 01:03:10,440 happen to produce some new pair of particles at rest. 993 01:03:10,440 --> 01:03:11,880 I would like to find the threshold 994 01:03:11,880 --> 01:03:14,290 for this particular creation process. 995 01:03:14,290 --> 01:03:14,790 OK? 996 01:03:14,790 --> 01:03:16,457 So you're going to play with one problem 997 01:03:16,457 --> 01:03:18,950 on the p-set that's kind of like that. 998 01:03:18,950 --> 01:03:20,630 Let's see, what do I have time to do? 999 01:03:20,630 --> 01:03:21,500 I think I will do-- 1000 01:03:24,655 --> 01:03:26,280 yeah, I think I can do two more things. 1001 01:03:36,010 --> 01:03:43,710 So all the dot products that I have been talking about so far 1002 01:03:43,710 --> 01:03:47,640 have been a dot product of a 4-vector with itself. 1003 01:03:47,640 --> 01:03:49,050 I did u dotted into u. 1004 01:03:49,050 --> 01:03:51,240 I did p dotted into p. 1005 01:03:51,240 --> 01:03:54,240 I invented a frame in which p has a particularly simple form. 1006 01:03:54,240 --> 01:03:56,198 And then when you actually do some of analysis, 1007 01:03:56,198 --> 01:03:59,213 you would probably take that p tote and dot it into itself. 1008 01:03:59,213 --> 01:04:00,630 I haven't done anything that looks 1009 01:04:00,630 --> 01:04:03,570 at the crossing between these two things, 1010 01:04:03,570 --> 01:04:07,435 dotting one into the other. 1011 01:04:07,435 --> 01:04:11,450 So let me to go through a very useful result that 1012 01:04:11,450 --> 01:04:14,570 follows by combining p with u. 1013 01:04:14,570 --> 01:04:16,730 There's a very specific notion of p 1014 01:04:16,730 --> 01:04:18,140 and a very specific notion of u. 1015 01:04:40,340 --> 01:04:44,900 Let's let p be the 4-momentum of a particle. 1016 01:04:44,900 --> 01:04:45,470 I call it a. 1017 01:04:58,780 --> 01:05:13,190 Let's let u be the 4-velocity not of a, but the 4-velocity 1018 01:05:13,190 --> 01:05:22,902 of observer O. So particle a might 1019 01:05:22,902 --> 01:05:24,860 be a muon that was created the upper atmosphere 1020 01:05:24,860 --> 01:05:26,960 and is crashing through our room right now. 1021 01:05:26,960 --> 01:05:29,780 Observer O might be your hyperactive friend 1022 01:05:29,780 --> 01:05:33,245 who is jogging through the room at half the speed of light. 1023 01:05:33,245 --> 01:05:39,530 The question I want to ask is, what does O measure 1024 01:05:39,530 --> 01:05:45,705 as the energy of particle a. 1025 01:06:11,630 --> 01:06:14,060 So the naive way to do this, which I will emphasize 1026 01:06:14,060 --> 01:06:18,320 is not wrong, what you might do is sort of go, 1027 01:06:18,320 --> 01:06:20,210 OK, well, we're sitting here. 1028 01:06:20,210 --> 01:06:22,140 This room is our laboratory. 1029 01:06:22,140 --> 01:06:23,910 I've measure this thing in my lab. 1030 01:06:23,910 --> 01:06:26,480 So I know p as I measure it. 1031 01:06:26,480 --> 01:06:28,280 I can see O jogging by. 1032 01:06:28,280 --> 01:06:32,180 So I know O's 4-velocity as I measure it. 1033 01:06:32,180 --> 01:06:35,390 So what I should do is figure out the Lorenz transformation 1034 01:06:35,390 --> 01:06:38,720 that takes me into the rest frame of O. 1035 01:06:38,720 --> 01:06:40,658 Once I have that Lorenz transformation, 1036 01:06:40,658 --> 01:06:42,200 I'll apply that Lorenz transformation 1037 01:06:42,200 --> 01:06:44,510 to the 4-vector p. 1038 01:06:44,510 --> 01:06:47,030 Boom, that will give me that energy. 1039 01:06:47,030 --> 01:06:47,960 That will work. 1040 01:06:47,960 --> 01:06:49,070 That will absolutely work. 1041 01:06:52,652 --> 01:06:54,110 But there's an easier way to do it. 1042 01:06:57,760 --> 01:07:11,010 So one thing you should note is that everybody represents that 1043 01:07:11,010 --> 01:07:17,636 4-velocity as an energy-- 1044 01:07:17,636 --> 01:07:19,760 excuse me, they represent the 4-momentum 1045 01:07:19,760 --> 01:07:21,573 as an energy and a 3-momentum. 1046 01:07:21,573 --> 01:07:23,240 In particular, though, they represent it 1047 01:07:23,240 --> 01:07:26,030 as the energy that they would measure 1048 01:07:26,030 --> 01:07:30,440 and the 3-momentum that they would measure. 1049 01:07:30,440 --> 01:07:39,730 So that means p as seen by O is e according to O 1050 01:07:39,730 --> 01:07:48,545 and p according to O. That are acquainted o is what we want. 1051 01:07:48,545 --> 01:07:49,920 And I just told you a moment ago, 1052 01:07:49,920 --> 01:07:52,208 you know, if you have p in your own reference frame, 1053 01:07:52,208 --> 01:07:54,000 and you have u in your own reference frame, 1054 01:07:54,000 --> 01:07:56,250 you can do this whole math with Lorenz transformations 1055 01:07:56,250 --> 01:07:57,000 and get it out. 1056 01:07:57,000 --> 01:08:05,930 But you also know that in O's own reference frame, 1057 01:08:05,930 --> 01:08:15,770 O represents their 4-velocity as 1 in the timelike direction, 0 1058 01:08:15,770 --> 01:08:17,135 in the spatial direction. 1059 01:08:25,550 --> 01:08:31,420 So what this means is if I go into O's reference frame, 1060 01:08:31,420 --> 01:08:42,140 if I go into their inertial reference frame, 1061 01:08:42,140 --> 01:08:45,710 notice that if I take the dot product of p and u, 1062 01:08:45,710 --> 01:08:49,275 I get e times 1 and p times 0. 1063 01:09:00,130 --> 01:09:02,439 So that is just negative. 1064 01:09:02,439 --> 01:09:05,207 It's exactly what I want-- modulo minus sign. 1065 01:09:05,207 --> 01:09:07,540 And so you go, OK, well, I'll flip my minus sign around. 1066 01:09:13,359 --> 01:09:15,069 And you think, OK, great, but I did 1067 01:09:15,069 --> 01:09:17,170 that using those quantities as written down 1068 01:09:17,170 --> 01:09:18,640 in O's reference frame. 1069 01:09:18,640 --> 01:09:20,680 And then you go, holy crap, that's 1070 01:09:20,680 --> 01:09:23,109 the invariant scalar product. 1071 01:09:23,109 --> 01:09:23,810 I'm done. 1072 01:09:23,810 --> 01:09:24,310 Mic drop. 1073 01:09:24,310 --> 01:09:26,080 Leave the room. 1074 01:09:26,080 --> 01:09:31,000 What this means is you start with p as you measure it, 1075 01:09:31,000 --> 01:09:33,399 u as you measure it. 1076 01:09:33,399 --> 01:09:35,680 Take the scalar product between the two of them. 1077 01:09:35,680 --> 01:09:37,840 Boom, the answer you want pops out. 1078 01:09:37,840 --> 01:09:39,910 No nonsense of Lorenz transformations. 1079 01:09:39,910 --> 01:09:42,194 None of that garbage needs to happen. 1080 01:09:42,194 --> 01:09:44,319 You just take that inner product and you've got it. 1081 01:10:27,760 --> 01:10:29,970 So that sort of says in words, no matter 1082 01:10:29,970 --> 01:10:32,565 what representation you choose to write down p and u in, 1083 01:10:32,565 --> 01:10:34,440 take the dot product between the two of them, 1084 01:10:34,440 --> 01:10:37,500 throw in a minus sign, you've got the energy of the particle 1085 01:10:37,500 --> 01:10:39,840 with p as measured by the observer with u. 1086 01:10:44,660 --> 01:10:47,810 It's sort of late in the hour, or the hour and a half 1087 01:10:47,810 --> 01:10:48,930 I should say. 1088 01:10:48,930 --> 01:10:51,060 So let me just sort of emphasize, 1089 01:10:51,060 --> 01:10:54,080 there are occasional moments in this class where 1090 01:10:54,080 --> 01:10:58,130 if you're dozing off a little bit, I suggest you pop up 1091 01:10:58,130 --> 01:11:00,800 and tattoo this into a neuron somewhere. 1092 01:11:00,800 --> 01:11:02,090 This is one of those moments. 1093 01:11:02,090 --> 01:11:02,330 OK? 1094 01:11:02,330 --> 01:11:04,670 This is a result that we're going to use over and over 1095 01:11:04,670 --> 01:11:06,380 and over again, because this holds-- 1096 01:11:06,380 --> 01:11:08,840 this isn't just in special relativity. 1097 01:11:08,840 --> 01:11:12,230 When we start talking about the behavior of things 1098 01:11:12,230 --> 01:11:14,120 near black holes, there's going to a place 1099 01:11:14,120 --> 01:11:15,150 where I basically at that point to say, 1100 01:11:15,150 --> 01:11:17,567 well, I'm going to use the fact that the observer measures 1101 01:11:17,567 --> 01:11:20,150 an energy that is given by-- and I'm going to write down that. 1102 01:11:20,150 --> 01:11:22,317 The dot product that's involved is a little bit more 1103 01:11:22,317 --> 01:11:24,170 complicated, because my metric is hairier. 1104 01:11:24,170 --> 01:11:26,470 But it's the exact same physical concept. 1105 01:11:26,470 --> 01:11:28,130 OK? 1106 01:11:28,130 --> 01:11:29,720 Let me just do one more. 1107 01:11:29,720 --> 01:11:32,260 And then I'll go talk, without getting into the math, 1108 01:11:32,260 --> 01:11:35,900 about what I will start with on Tuesday. 1109 01:11:35,900 --> 01:11:38,810 Last 4-velocity, which is probably useful for us 1110 01:11:38,810 --> 01:11:49,795 to quickly talk about is-- 1111 01:11:52,330 --> 01:11:56,170 so we've talked a lot about 4-velocity. 1112 01:11:56,170 --> 01:11:57,952 That is just one piece-- when we're 1113 01:11:57,952 --> 01:12:00,160 talking about sort of the kinematics of bodies moving 1114 01:12:00,160 --> 01:12:02,290 in spacetime, you need more information than just velocity. 1115 01:12:02,290 --> 01:12:03,748 Sometimes things are moving around. 1116 01:12:03,748 --> 01:12:06,980 There's additional forces acting on them. 1117 01:12:06,980 --> 01:12:10,540 And so we also care about the 4-acceleration. 1118 01:12:26,530 --> 01:12:29,300 And so this is what I get when I take 1119 01:12:29,300 --> 01:12:34,520 the derivative with respect to proper time of the 4-velocity. 1120 01:12:34,520 --> 01:12:37,700 So will there be some homework exercises that use this. 1121 01:12:37,700 --> 01:12:40,340 The main thing which I want to emphasize to sort of conclude 1122 01:12:40,340 --> 01:12:43,610 our calculations for today is that when 1123 01:12:43,610 --> 01:12:46,850 I talk about a 4-velocity of acceleration, 1124 01:12:46,850 --> 01:12:49,820 this has an extremely important property. 1125 01:12:49,820 --> 01:13:02,060 It is always the case that a dotted into u equals 0. 1126 01:13:07,220 --> 01:13:09,650 If you're used to sort of 3-dimensional intuition, 1127 01:13:09,650 --> 01:13:11,210 that may seem weird. 1128 01:13:11,210 --> 01:13:12,230 OK? 1129 01:13:12,230 --> 01:13:15,760 Anytime you see a car accelerate from a stop, that's 1130 01:13:15,760 --> 01:13:18,620 a case in which its acceleration is clearly not 1131 01:13:18,620 --> 01:13:21,110 orthogonal to its velocity. 1132 01:13:21,110 --> 01:13:24,710 But the issue here is, this is not a spatial dot product. 1133 01:13:24,710 --> 01:13:26,750 This is a space time dot product. 1134 01:13:26,750 --> 01:13:28,880 And some of your intuition has to go out the window 1135 01:13:28,880 --> 01:13:29,830 because of that. 1136 01:13:29,830 --> 01:13:32,090 It's very simple to prove this. 1137 01:13:32,090 --> 01:13:37,430 Remember, u dot u equals minus 1. 1138 01:13:37,430 --> 01:13:46,110 So d of u dot u, d tau, which is just 2, u 1139 01:13:46,110 --> 01:13:53,120 dot a is the derivative of minus 1, which is 0. 1140 01:13:53,120 --> 01:13:53,880 OK? 1141 01:13:53,880 --> 01:13:56,320 So this is something that we will exploit. 1142 01:13:56,320 --> 01:14:00,210 If you want to describe the relativistic kinetics 1143 01:14:00,210 --> 01:14:04,140 of an accelerating body, this is a great thing 1144 01:14:04,140 --> 01:14:05,565 that we can use to exploit. 1145 01:14:05,565 --> 01:14:07,440 You often need a little bit more information. 1146 01:14:07,440 --> 01:14:09,900 We have to give you as a bit of additional information 1147 01:14:09,900 --> 01:14:12,442 some knowledge about what the orientation of the acceleration 1148 01:14:12,442 --> 01:14:13,905 is and things like that. 1149 01:14:13,905 --> 01:14:15,030 So whenever you are given-- 1150 01:14:15,030 --> 01:14:16,290 I'll get to you in just a sec-- whenever you're 1151 01:14:16,290 --> 01:14:18,780 given any kind of differential quantity like this, 1152 01:14:18,780 --> 01:14:21,330 it's not enough to know-- it's like the acceleration of loss, 1153 01:14:21,330 --> 01:14:23,080 you have to also have boundary conditions. 1154 01:14:23,080 --> 01:14:26,110 And that sort of tells you what the initial direction is. 1155 01:14:26,110 --> 01:14:26,610 Question? 1156 01:14:26,610 --> 01:14:29,040 AUDIENCE: Is that time still the proper time? 1157 01:14:29,040 --> 01:14:31,050 SCOTT HUGHES: That time is the proper time, yes. 1158 01:14:31,050 --> 01:14:32,410 AUDIENCE: So it's not accelerating? 1159 01:14:32,410 --> 01:14:33,400 SCOTT HUGHES: That's right. 1160 01:14:33,400 --> 01:14:35,880 So you can still define a proper time for an accelerating 1161 01:14:35,880 --> 01:14:36,420 observer. 1162 01:14:36,420 --> 01:14:37,605 It will not relate-- 1163 01:14:40,130 --> 01:14:40,880 hold that thought. 1164 01:14:40,880 --> 01:14:41,960 You're going to play this a little bit more 1165 01:14:41,960 --> 01:14:43,070 in a future problem set. 1166 01:14:43,070 --> 01:14:45,690 I mean, the key thing is that the way-- 1167 01:14:45,690 --> 01:14:49,310 if you have an d observer, an interval of proper time 1168 01:14:49,310 --> 01:14:51,950 as compared to an interval of time for, you know, someone 1169 01:14:51,950 --> 01:14:55,670 in a rest frame that sees this person accelerate away, 1170 01:14:55,670 --> 01:14:58,020 the conversion between the intervals of time, 1171 01:14:58,020 --> 01:14:59,260 the two measure, it evolves. 1172 01:15:02,050 --> 01:15:05,540 So, you know, let's say you're in this room with me. 1173 01:15:05,540 --> 01:15:08,680 In fact, it turns out that if you accelerate 1174 01:15:08,680 --> 01:15:12,100 at 1G for a year, you get to very 1175 01:15:12,100 --> 01:15:13,960 close to the speed of light. 1176 01:15:13,960 --> 01:15:15,910 So let's say that you were in a rocket ship 1177 01:15:15,910 --> 01:15:19,780 right now that launched with an acceleration of G. Initially, 1178 01:15:19,780 --> 01:15:22,990 you and I synchronize our watches. 1179 01:15:22,990 --> 01:15:24,840 And so an interval of a second to me 1180 01:15:24,840 --> 01:15:26,560 is the same as a second to you. 1181 01:15:26,560 --> 01:15:28,742 Half a year later, you're moving at something 1182 01:15:28,742 --> 01:15:29,950 like half the speed of light. 1183 01:15:29,950 --> 01:15:33,490 And I will see a noticeable time delay. 1184 01:15:33,490 --> 01:15:35,290 An interval of a second as you measure it 1185 01:15:35,290 --> 01:15:38,415 looks long compared to me. 1186 01:15:38,415 --> 01:15:39,790 Six months later, you're actually 1187 01:15:39,790 --> 01:15:40,870 quite close to the speed of light, 1188 01:15:40,870 --> 01:15:42,120 and it gets dilated even more. 1189 01:15:44,650 --> 01:15:46,270 So last thing, which I'm going to say, 1190 01:15:46,270 --> 01:15:49,570 and I'm not going to get into too much detail with this yet, 1191 01:15:49,570 --> 01:15:56,400 is we're going to begin next time by making 1192 01:15:56,400 --> 01:16:00,120 a little bit more formal some of the notions that go around. 1193 01:16:00,120 --> 01:16:03,420 So we've increased some physics and some vectors. 1194 01:16:03,420 --> 01:16:10,120 I've given you guys one tensor so far, the metric tensor. 1195 01:16:16,025 --> 01:16:17,650 And so I'm going to give you-- in fact, 1196 01:16:17,650 --> 01:16:22,068 I will write down a very precise definition of this right now, 1197 01:16:22,068 --> 01:16:23,860 and we'll pick it up from there on Tuesday. 1198 01:16:23,860 --> 01:16:25,843 So the basic idea-- so you guys have see-- 1199 01:16:25,843 --> 01:16:28,135 the one tensor you've seen so far is the metric tensor. 1200 01:16:28,135 --> 01:16:31,570 And what the metric is is it's sort of a mathematical object 1201 01:16:31,570 --> 01:16:34,720 that I put in a pair of 4-vectors. 1202 01:16:34,720 --> 01:16:38,410 And it spits out a quantity that is a Lorenz invariant 1203 01:16:38,410 --> 01:16:42,040 scalar that characterizes what we call the inner product 1204 01:16:42,040 --> 01:16:44,680 of those two 4-vectors. 1205 01:16:44,680 --> 01:16:47,080 More generally, I'm going to define 1206 01:16:47,080 --> 01:17:12,110 a tensor of type 0 N as a function or mapping of N 1207 01:17:12,110 --> 01:17:34,920 4-vectors into Lorenz invariant scalars, which is linear 1208 01:17:34,920 --> 01:17:37,664 in each of its N arguments. 1209 01:17:50,510 --> 01:17:52,360 So I will pick it up here on Tuesday. 1210 01:17:52,360 --> 01:17:58,160 And let me just say in words, the metric is a 0 2 tensor. 1211 01:17:58,160 --> 01:18:00,770 I put in two 4-vectors. 1212 01:18:00,770 --> 01:18:03,860 It spits out a Lorenz invariant scalar. 1213 01:18:03,860 --> 01:18:06,770 We're going to before too long come up 1214 01:18:06,770 --> 01:18:11,120 with a couple of things that involve three real vectors-- 1215 01:18:11,120 --> 01:18:14,930 excuse me three 4-vectors, too many numbers here, 1216 01:18:14,930 --> 01:18:18,170 a trio of 4-vectors, which it then maps to a Lorenz invariant 1217 01:18:18,170 --> 01:18:19,010 scalar. 1218 01:18:19,010 --> 01:18:21,770 Some of them will take in four 4-vectors 1219 01:18:21,770 --> 01:18:23,900 and produce a Lorenz invariant scalar. 1220 01:18:23,900 --> 01:18:25,940 Notice I wrote this in sort of funny way. 1221 01:18:25,940 --> 01:18:30,100 The 0 N sort of begs for there to be sort of an N 0. 1222 01:18:30,100 --> 01:18:30,950 OK? 1223 01:18:30,950 --> 01:18:34,010 To do that I have to introduce an object that 1224 01:18:34,010 --> 01:18:35,718 is sort of dual to a vector. 1225 01:18:35,718 --> 01:18:37,010 We're going to talk about that. 1226 01:18:37,010 --> 01:18:39,080 Those are objects called one-forms, 1227 01:18:39,080 --> 01:18:41,450 which actually happen to be a species of vector. 1228 01:18:41,450 --> 01:18:44,210 We're actually going to then learn that the vector is itself 1229 01:18:44,210 --> 01:18:45,590 a tensor. 1230 01:18:45,590 --> 01:18:48,343 And so we will make a very general classification 1231 01:18:48,343 --> 01:18:49,010 of these things. 1232 01:18:49,010 --> 01:18:52,190 And we'll see that vectors are just a subset of these tensors. 1233 01:18:52,190 --> 01:18:55,340 And at last, we'll sort of have all the mathematics in place. 1234 01:18:55,340 --> 01:18:57,380 We can sort of lose some of these distinctions 1235 01:18:57,380 --> 01:18:59,810 and just life goes on, and we can start actually 1236 01:18:59,810 --> 01:19:01,220 doing some physics with these. 1237 01:19:01,220 --> 01:19:04,330 All right, I will pick it up there on Tuesday.